A Vertex-Weighted Tutte Symmetric Function, and Constructing Graphs with Equal Chromatic Symmetric Function

This paper has two main parts. First, we consider the Tutte symmetric function $XB$, a generalization of the chromatic symmetric function. We introduce a vertex-weighted version of $XB$ and show that this function admits a deletion-contraction relation. We also demonstrate that the vertex-weighted $XB$ admits spanning-tree and spanning-forest expansions generalizing those of the Tutte polynomial by connecting $XB$ to other graph functions. Second, we give several methods for constructing nonisomorphic graphs with equal chromatic and Tutte symmetric functions, and use them to provide specific examples.


Introduction
The chromatic symmetric function X G of a graph G, introduced by Stanley in the 1990s [42], is an extension of the chromatic polynomial that (among other things) counts for each integer partition λ = (λ 1 , . . . , λ k ) the number of partitions of V (G) into stable sets of sizes λ 1 , . . . , λ k . This function has seen a recent resurgence of interest, including research focusing on the expansion of X G in the bases of elementary symmetric functions [9,15,16,23] and Schur functions [37,38,45], and the conjecture that X G distinguishes nonisomorphic trees [3,26].
Other results have extended the definition of X G in various ways to include representation theoretic and graph theoretic considerations, including chromatic quasisymmetric functions [1,22,41] and chromatic symmetric functions in noncommuting variables [17,24].
In [13], the second and third authors extended the chromatic symmetric function to graphs G equipped with vertex weights in the form of a function w : V (G) → N. The extended function X (G,w) satisfies a natural deletion-contraction relation, which can be used to extend identities of X G to this broader class of graphs, and prove new results.
In this paper, we continue the work of [13] in multiple ways. Following an exposition in Section 2 of necessary background on graphs and symmetric functions, in Section 3 we extend the function X (G,w) to include Stanley's Tutte symmetric function [43], which is a natural extension of the Tutte polynomial. We show that the resulting function XB (G,w) (t, x 1 , x 2 , . . . ) of t and variables x 1 , x 2 , . . . satisfies an edge deletion-contraction relation generalizing that of X (G,w) .
In Section 4 we show that XB (G,w) is a specialization of the V -polynomial of Ellis-Monaghan and Moffatt [19], and is thus closely related to many other graph functions. For example, we demonstrate that up to a change of variables the vertex-weighted version of XB is equivalent to the W -polynomial of Noble and Welsh [35] by showing that the two functions satisfy the same base cases and recurrence relations, providing a strengthening Noble and Welsh's analogous result on the equivalence of unweighted XB and the unweighted Wpolynomial (or U-polynomial). We similarly show that the vertex-weighted XB is equivalent to a natural vertex-weighted extension of the polychromate of Brylawski [7] and the weighted (r, q)-chromatic function of Klazar, Loebl, and Moffatt [27], generalizing proofs of equivalence in unweighted graphs made in the aforementioned references as well as the work of Merino and Noble [33] and Sarmiento [40].
In Section 5 we use the relationship between XB and the V -polynomial to derive spanningtree and spanning-forest expansions for XB. We show that the spanning-tree expansion specializes to a well-known analogous expansion of the Tutte polynomial, and is a natural improvement of the p-basis expansion formula for XB originally introduced in [10]. From this formula we likewise derive an expansion of the chromatic symmetric function in terms of spanning trees with no external activity, and show that this expansion refines its classical p-basis expansion [42].
In Section 6, we use deletion-contraction relations to provide multiple original methods for constructing pairs of nonisomorphic graphs with equal X G and/or XB G . In particular, we demonstrate graph pairs with arbitrarily high girth whose Tutte symmetric functions agree.
In Section 7, we further use the equivalence between XB G and the W -polynomial of G to find additional families of vertex-weighted graphs with the same XB, and in particular we show how to construct arbitrarily large vertex-weighted paths with equal XB (similar results are found in the independent work [2] by Aliniaeifard, Wang, and van Willigenburg).
We conclude in Section 8 with further directions and conjectures. We note how the examples in Sections 6 and 7 suggest new lines of research related to open problems regarding the chromatic symmetric function, particularly the conjecture that the chromatic symmetric function distinguishes nonisomorphic trees.

Fundamentals of Symmetric Functions and Graphs
An integer partition (or just partition) is a tuple λ = (λ 1 , . . . , λ k ) of positive integers such that λ 1 ≥ · · · ≥ λ k . The integers λ i are the parts of λ. If k i=1 λ i = n, we say that λ is a partition of n, and we write λ ⊢ n, or |λ| = n. The number of parts k is the length of λ, and is denoted by l(λ). The number of parts equal to i in λ is given by r i (λ). (2) , . . . ) for every permutation σ of the positive integers N. The algebra of symmetric functions Λ is the subalgebra of R[[x 1 , x 2 , . . . ]] consisting of those symmetric functions f that are of bounded degree (that is, there exists a positive integer n such that every monomial of f has degree ≤ n). Furthermore, Λ is a graded algebra, with natural grading where Λ d consists of symmetric functions that are homogeneous of degree d [29,44].
Each Λ d is a finite-dimensional vector space over R, with dimension equal to the number of partitions of d (and thus, Λ is an infinite-dimensional vector space over R). Some commonlyused bases of Λ that are indexed by partitions λ = (λ 1 , . . . , λ k ) include: • The monomial symmetric functions m λ , defined as the sum of all distinct monomials of the form • The power-sum symmetric functions, defined by the equations • The elementary symmetric functions, defined by the equations We also make use of the augmented monomial symmetric functions, defined by Given a symmetric function f and a basis b of Λ, we say that f is b-positive if when we write f in the basis b, all coefficients are nonnegative.
We define the symmetric function involution ω by ω(p λ ) = (−1) |λ|−l(λ) p λ . A graph G = (V, E) consists of a vertex set V and an edge multiset E where the elements of E are (unordered) pairs of (not necessarily distinct) elements of V . An edge e ∈ E that contains the same vertex twice is called a loop. If there are two or more edges that each contain the same two vertices, they are called multi-edges. A simple graph is a graph G = (V, E) in which E does not contain loops or multi-edges (thus, E ⊆ V 2 ). If {v 1 , v 2 } is an edge, we will write it as v 1 v 2 = v 2 v 1 . The vertices v 1 and v 2 are the endpoints of the edge v 1 v 2 . We will use V (G) and E(G) to denote the vertex set and edge multiset of a graph G, respectively.
Two graphs G and H are said to be isomorphic if there exists a bijective map f : The complement of a simple graph G = (V, E) is denoted G, and is defined as G = (V, V 2 \E), so in G every edge of G is replaced by a nonedge, and every nonedge is replaced by an edge.
A subgraph of a graph G is a graph The vertices v 1 and v k are the endpoints of the path. A cycle in a graph is a nonempty sequence of distinct edges Note that in a simple graph every cycle must have at least 3 edges, although in a nonsimple graph there may be cycles of size 1 (a loop) or 2 (multi-edges).
A graph G is connected if for every pair of vertices v 1 and v 2 of G there is a path in G with v 1 and v 2 as its endpoints. The connected components of G are the maximal induced subgraphs of G which are connected. The number of connected components of G will be denoted by c(G).
The complete graph K n on n vertices is the unique simple graph having all possible edges, that is, . Given a graph G, there are two commonly used operations that produce new graphs. One is deletion: given an edge e ∈ E(G), the graph of G with e deleted is the graph G ′ = (V (G), E(G)\{e}), and is denoted G\e. Likewise, if S is a multiset of edges, we use G\S to denote the graph (V (G), E(G)\S).
The other operation is the contraction of an edge e = v 1 v 2 , denoted G/e. If v 1 = v 2 (e is a loop), we define G/e = G\e. Otherwise, we create a new vertex v * , and define G/e as the graph is the set of edges with at least one of v 1 or v 2 as an endpoint, and E(v * ) consists of each edge in E(v 1 , v 2 )\e with the endpoint v 1 and/or v 2 replaced with the new vertex v * . Note that this is an operation on a (possibly nonsimple) graph that identifies two vertices while keeping and/or creating multi-edges and loops.
There is also a different version of edge contraction that is defined only on simple graphs. In the case that G is a simple graph, we define the simple contraction G ∤ e to be the same as G/e except that after performing the contraction operation, we delete any loops and all but a single copy of each multi-edge so that the result is again a simple graph.
Let G = (V (G), E(G)) be a (not necessarily simple) graph. A map κ : The chromatic symmetric function X G of G is defined as where the sum runs over all proper colorings κ of G. Note that if G contains a loop then X G = 0, and X G is unchanged by replacing each multi-edge by a single edge.

Vertex-Weighted Graphs and their Colorings
A vertex-weighted graph (G, w) consists of a graph G and a weight function w : V (G) → N.
Given two vertex-weighted graphs (G 1 , w 1 ) and (G 2 , w 2 ), we call a map f : Given a vertex-weighted graph (G, w) and a non-loop edge e = v 1 v 2 ∈ E(G) we define its contraction by e to be the graph (G/e, w/e), where w/e is the weight function such that (w/e)(v) = w(v) if v is the not the contracted vertex v * , and (w/e)(v * ) = w(v 1 ) + w(v 2 ) (if e is a loop, we define the contraction of (G, w) by e to be (G\e, w)).
In [13], the authors extended X G to vertex-weighted graphs as where again the sum ranges over all proper colorings κ of G. In this setting the chromatic symmetric function admits the deletion-contraction relation [13] as well as the version using simple contraction in the case that G is simple. Note also that if two vertex-weighted graphs are w-isomorphic, then they must have the same chromatic symmetric function. The converse is not true even in the unweighted case [42].

The Weighted Version of the Tutte Symmetric Function
In this section, we extend the definition of the vertex-weighted chromatic symmetric function to include all colorings of a graph, not just the proper ones. To this end, for a given (not necessarily proper) coloring κ of G, we define . We then define the Tutte symmetric function of a vertex-weighted graph 1 (G, w) to be the following analogue of the Tutte symmetric function introduced by Stanley in [43]: where the sum is over all colorings κ of G (not just the proper ones). This name comes from the fact that the original function admits the Tutte polynomial T G (x, y) as a specialization via the relation where c(G) is the number of connected components of G. Given a partition π of V (G) (into nonempty blocks), let e(π) be the number of edges of G whose endpoints lie in the same block of π, and λ(π) the partition whose parts are the total weights of the blocks of π. We may verify the following lemma, an extension of the corresponding result on unweighted graphs: Proof. For λ = (λ 1 , . . . , λ k ) it suffices to show that the coefficient of From the defining equation (3) it is clear that we may only get the monomial x λ 1 1 . . . x λ k k by choosing a coloring κ of G with vertices of total weight λ i receiving the color i for each i, and then giving it the coefficient (1 + t) cκ(G) . Each such coloring κ corresponds to a partition 1 The function is also known as the bad-coloring chromatic symmetric function, hence the notation XB. We continue using XB as it is more common in the literature and less confusing in this context than the original X G (x; t).
π ⊢ V (G) into k blocks where the i th block consists of vertices colored i by κ, and these receive a coefficient of (1 + t) e(π) since the monochromatic edges are exactly those that are entirely contained within a block of π.
Conversely, each π ⊢ V (G) contributes (with coefficient (1 + t) e(π) ) exactly ( ∞ i=1 r i (λ)!) colorings with monomial x λ 1 1 . . . x λ k k : the one where the i th block of π gets color i, permuting the color choices among blocks of the same total weight, and the conclusion follows.
We use the convention 0 0 = 1, so that when t = −1 we have On vertex-weighted graphs, XB (G,w) admits the following deletion-contraction relation that generalizes the deletion-contraction relation of [13]: Proof. First, note that when t = −1, the deletion-contraction relation (6) reduces to (1), so we may assume t = −1. Furthermore, the case when e is a loop follows immediately from the definition of XB, so we may assume that e is not a loop.
Let v 1 and v 2 be the endpoints of e. We start with the right-hand side of (6) and expand using the definition ( Note that all colorings of G are also colorings of G\e, and vice versa. We split the κ in the first summand based on κ(v 1 ) and κ(v 2 ). In those κ where κ(v 1 ) = κ(v 2 ), we have c κ (G\e) = c κ (G), so x κ (G\e, w, t) = x κ (G, w, t). In all κ with κ(v 1 ) = κ(v 2 ), we have c κ (G\e) = c κ (G) − 1 because of the missing edge e, so for these κ, we have x κ (G\e, w, t) = (1 + t) −1 x κ (G, w, t).
For the second summand, note that every coloring κ of G/e corresponds naturally to a coloring κ of G with κ(v 1 ) = κ(v 2 ), and vice-versa (we will use the same κ to denote both of these colorings in a slight abuse of notation). For these κ we will have c κ (G/e) = c κ (G) − 1 since we are missing the contracted edge e, and thus for each such κ we will have x κ (G/e, w/e, t) = (1 + t) −1 x κ (G, w, t). Putting everything together, we have As a consequence of this relation, we can derive a p-basis expansion formula by simply replacing (−1)s with ts in ( [13], Lemma 3) to give the following analogue of the original formula in [43]: where λ(G, w, S) is the integer partition whose parts are the total weights of the connected components of ((V (G), S), w).

Relating the Tutte Symmetric Function With Other Graph Functions
Note that the deletion-contraction relation (6) together with XB (G,w) = p (w 1 ,...,w k ) when (G, w) is a graph with no edges and vertices of weights w 1 ≥ · · · ≥ w k can be taken as an alternative definition of XB (G,w) . This formulation is closely related to the more general V -polynomial, defined as a function • J is a torsion-free commutative semigroup (e.g. (N, +) or (2 N , ∩)); • The function uses a set of commuting indeterminates x j indexed by elements j ∈ J, and a set of commuting indeterminates γ e indexed by edges e ∈ E(G).
For brevity, in what follows we will often fix J and the variables x j , and consider V as a function V (G,w) on vertex-weighted graphs. Given these inputs, the V -polynomial is defined by the following relations [19]: • If (G, w) is a graph with no edges and vertices of weights It may be shown from these relations that the V -polynomial satisfies [19] where C G (S) is the set of connected components of the graph (V (G), S), and for c ∈ C G (S), |c| is the sum (using the operation of J) of the weights of the vertices in c.
Using either the recurrence relations or the expansion (8), we may verify that XB (G,w) is a special case of the V -polynomial in which J = (N, +), γ e = t for all e ∈ E(G), and each variable x n is replaced by the power-sum symmetric function p n (x 1 , x 2 , . . . ). That is, The function XB (G,w) is also closely related to other specializations of the V -polynomial. A notable example is the W -polynomial from invariant theory [35] (and its unweighted version, the U-polynomial), which has been studied both in its own right [34] and in relation to the chromatic symmetric function [3,4]. This (nonsymmetric) function W (G,w) (y, x 1 , x 2 , . . . ) on vertex-weighted graphs is defined by the following relations: • If (G, w) is a graph with no edges and vertices of weights w 1 ≥ · · · ≥ w k , we have Note that if J = (N, +) and γ e = y − 1 for all e ∈ E(G), then so the W -polynomial may be derived from the V -polynomial [19]. One can prove either by specializing (8) or induction on the number of edges as in [35] that the W -polynomial satisfies where c 1 , . . . , c k are the total weights of the connected components of the vertex-weighted graph ((V (G), S), w).
We say that two functions on vertex-weighted graphs (G, w) are equivalent if given one, we can entirely recover the other, without knowing the graph (G, w). We show the following generalization of ( [35], Theorem 6.2): Proof. We actually prove a stronger statement, that given W (G,w) , we may recover the p-basis expansion of XB (G,w) via the substitution and conversely, given the p-basis expansion of XB (G,w) , we may recover W (G,w) by dividing by t |V (G)| , setting t = y − 1, and replacing each p k with tx k . This stronger statement may be proven as a simple vertex-weighted generalization of the argument from ( [35], Theorem 6.2) by showing that these substitutions take (7) to (11) and vice-versa.
We provide a different proof by showing that this substitution works not just for these equations, but for the base cases and inductive steps of the recursive definitions for XB (G,w) and W (G,w) . In this sense these functions are not only equivalent, but essentially the same up to a change of variables.
The base cases for both functions are vertex-weighted graphs with no edges. Let (G, w) be a vertex-weighted graph with no edges and vertices of weights w 1 ≥ · · · ≥ w k . Then . . x w k , and we now verify that the substitution works. Going from W to XB we have: and the converse is analogous. For the inductive step, assume that we have demonstrated that this substitution is valid for graphs with at most m edges for some m. Let (G, w) be a vertex-weighted graph with m + 1 edges and let e be an edge of G. Starting with the W -polynomial and using deletioncontraction we have two cases. First, if e is a loop, then W (G,w) = yW (G\e,w) . Then applying our substitution we may derive (t + 1)XB (G\e,w) = XB (G,w) , and the converse is analogous.
If e is not a loop, then W (G,w) = W (G\e,w) + W (G/e,w/e) (note that G\e and G/e have a different number of vertices). We make the substitution x i = p i t , y = t + 1, and multiply by t |V (G)| . Then by the inductive hypothesis the resulting function is XB (G\e,w) +tXB (G/e,w/e) = XB (G,w) as desired, and again the converse process of recovering W from XB is analogous.
The function XB (G,w) is also related to the weighted (r, q)-chromatic function of [27]. For a vertex-weighted graph (G, w) with n vertices, this function is defined as where C(S) is the set of connected components of (V (G), S), and w(c) is the total weight of the component c.
This function has a natural extension with an additional parameter in the form Note that from (13) it is clear that B (G,w) (and thus also M (G,w) ) may be derived from the V -polynomial by taking J = (N, +) and γ e = t for all e, and then substituting x n = n−1 i=0 r w(c)q i . Using the arguments from ( [27], Section 3) and adjusting them to the vertex-weighted case it is easy to show that Finally, we mention a closely related graph function that is not a specialization of the V -polynomial. Retaining the notation used for the m-basis expansion of XB given in (5), define the polychromate of a vertex-weighted graph (G, w) as where here letting λ = λ(π) we have x λ(π) = x λ 1 · · · x λ l(λ) . This is a vertex-weighted generalization of a function originally introduced by Brylawski [7]. Unfortunately, even in this vertex-weighted form, ν (G,w) is not a specialization of the V -polynomial, as when (G, w) has no edges and vertices of weights w 1 , . . . , w k we find that ν (G,w) is equal to the sum of x w 1 . . . x w k and all of the x λ where λ is a coarsening of the partition (w 1 , . . . , w k ). Even modifying the weight set J does not give a reasonable fix to this problem.
Nonetheless, we may easily see by comparing (14) with (5) that the vertex-weighted polychromate is equivalent to the vertex-weighted XB and thus to the vertex-weighted (r, q)-chromatic function and the W -polynomial. This extends previously known results that showed the equivalence of these four functions on unweighted graphs [27,33,40]. A more thorough summary of these functions and their properties on unweighted graphs is given in [21].
It is reasonable to ask what advantages are introduced by using the vertex-weighted Tutte symmetric function as opposed to any of these equivalent graph functions. For one, the theory of symmetric function bases may be applied to find encoded information that is much more difficult to detect using the other functions. It is already known that the chromatic and Tutte symmetric functions on vertex-weighted graphs encode information, such as an enumeration of ordered pairs of acyclic orientations and certain maps on their sinks [13,14] or intersections of maximal stable sets [12], that to the best of the authors' knowledge have not been replicated by these other functions. Furthermore, as the Tutte polynomial is the universal graph polynomial with a deletion-contraction relation [18], it is natural to try to extend its properties to vertex-weighted graphs. In the next section we will derive further expansions of the Tutte symmetric function that naturally generalize classical expansions of the Tutte polynomial, lending solid evidence to the claim that the Tutte symmetric function is in some sense the natural vertex-weighted extension of the Tutte polynomial.

Spanning Tree and Spanning Forest Expansions for XB
Properties of the V -polynomial specialize naturally to properties of XB. In particular, by considering results in [31] we may derive spanning tree and spanning forest expansions for XB that are natural generalizations of well-known formulas for the Tutte polynomial. We will need the following definitions: A spanning forest of a graph G is an acyclic subgraph that contains all vertices of G. A spanning tree of G is a spanning forest of G with the same number of connected components as G. In what follows, we assume that the edges of G have been given some arbitrary total ordering. Given a fixed spanning tree T of G, we say that an edge f ∈ T is internally active with respect to T if it is the smallest edge in the set {e ∈ E(G) : (T \f ) ∪ e is a spanning tree}, and internally inactive with respect to T otherwise. An edge f / ∈ T is said to be externally active with respect to T if f is the smallest edge in the unique cycle of T ∪ f , and externally inactive with respect to T otherwise. We may also extend the notion of external activity to spanning forests F by defining an edge f / ∈ F to be externally inactive with respect to F if F ∪ f is acyclic, and otherwise applying the same definition as for trees.
Theorem 6 ([31], Theorems 5.1 and 6.2). Let (G, w) be a vertex-weighted graph with some arbitrary total order on its edges, and let T (G) be the set of spanning trees of G. For any T ∈ T (G), let ii(T ), ia(T ), ei(T ), ea(T ) denote respectively the number of internally inactive, internally active, externally inactive, and externally active edges of G with respect to T ∈ T (G). Also, let II(T ) be the set of internally inactive edges of G with respect to T . Then Additionally, let F (G) be the set of spanning forests of G. For F ∈ F (G), let |F | denote the number of edges of F , let ea(F ) denote the number of externally active edges of F , and let λ(F ) denote the partition whose parts are the total weights of the connected components of F . Then The spanning forest expansion (16) was also already known in an equivalent form for the W -polynomial [35]. Upon taking t = −1 (and 0 0 = 1) this formula reduces to a vertexweighted generalization of the chromatic symmetric function analogue of Whitney's Broken Circuit Theorem ( [42], Theorem 2.9).
On the other hand, by using the substitution formula (4) between XB and the Tutte polynomial, we may verify that equation (15) is a direct generalization of the well-known formula which provides further strong justification for the choice of XB as the natural symmetric function analogue of T G .
Furthermore, retaining the total order on the edges in Theorem 5, consider the mapping M : 2 |E(G)| → T (G) defined as follows: 1. Input S ⊆ E(G) and the total order of E(G). We will output T , the edges of a spanning tree of G. We start with T = S. Conversely, for T ∈ T (G), let II(T ) be the set of internally inactive edges of T , and EI(T ) the set of externally inactive edges. Then it is easy to verify that M −1 (T ) will consist of those S ⊆ E(G) such that II(T ) ⊆ S and EI(T ) ∩ S = ∅ (and where S can contain any subset of the internally and externally active edges of T ). Thus, starting from the spanning tree formula (15), if for each T ∈ T (G) we expand (t + 1) ea(T ) and XB (T,w)/II(T ) using the p-basis expansion (7), the monomials of the form t i p λ will be exactly those monomials of the p-basis formula for the whole graph (G, w) coming from S ∈ M −1 (T ). In this manner, the spanning tree expansion (15) also represents a way to refine the full p-basis expansion of XB.
Furthermore, specializing at t = −1 yields which provides a spanning tree formula for the chromatic symmetric function. This is interesting in its own right, and will be discussed further in Section 8.

Constructing Graphs with Equal Chromatic and Tutte Symmetric Functions
As with any graph function, it is natural to consider the extent to which the chromatic symmetric function distinguishes nonisomorphic graphs. The original chromatic symmetric function X G fails to distinguish nonisomorphic graphs with as few as five vertices [42], and Orellana and Scott [36] used a modular relation on triangles to construct families of infinitely many pairs of graphs with equal chromatic symmetric function 2 .
The deletion-contraction relation on X (G,w) and XB (G,w) is a powerful and simple tool for finding such constructions. Throughout this section, we will repeatedly use the following simple but fundamental lemma: w 1 ) and (G 2 , w 2 ) be vertex-weighted graphs, and let e 1 ∈ E(G 1 ) and e 2 ∈ E(G 2 ) be edges.
Proof. These claims follow immediately from equation (2) and Lemma 2 respectively.
In particular both parts of Lemma 7 hold when their equalities are replaced by wisomorphism of the corresponding graphs. Note that in the case of XB we cannot replace contraction by simple contraction because unlike with the chromatic symmetric function, graphs that differ only by replacing edges with multi-edges or vice versa still have different XB.

Constructions with Split Graphs
A bipartite graph is a graph that has a proper 2-coloring, that is, a graph whose vertices may be partitioned into two stable sets. A split graph is a graph that arises from taking a simple bipartite graph G with V (G) partitioned into nonempty stable sets S 1 and S 2 , and switching all of the nonedges in either (but not both) of G| S 1 and G| S 2 to edges. Thus, the vertices of a split graph may be partitioned (not necessarily uniquely) into a stable set and a clique. The class of split graphs can also be characterized by the property that they contain no induced subgraph isomorphic to a five-vertex cycle, a four-vertex cycle, or the complement of a four-vertex cycle [25].
There is a natural way noted by Loebl and Sereni [28] to associate to any possibly non-simple (unweighted) graph a corresponding simple split graph: given a graph G = (V, E), with V (G) = {v 1 , . . . , v n } and E(G) = {e 1 , . . . , e m }. Then the split graph sp(G) corresponding to G has vertex set V (sp(G)) = {t 1 , t 2 , . . . , t n , t n+1 , . . . , t n+m }, and edge set E(sp(G)) = {t i t j : 1 ≤ i < j ≤ n} ∪ {t i t n+j , t i ′ t n+j : e j = v i v i ′ in G}. In other words, sp(G) is formed by taking the vertices of G, making them into a clique, and then adding a "hat" corresponding to each edge of G. Using the above notation, we say that vertex t n+j of sp(G) is the splitting vertex of the edge e j = v i v i ′ in G. The construction is illustrated in Figure 1. For any two nonisomorphic graphs of more than three vertices the corresponding split graphs are clearly nonisomorphic (since the largest clique of the split graph must correspond to the vertex set of the original graph), so distinguishing split graphs is (up to some processing) equivalent to distinguishing all graphs. This motivates considering which functions distinguish split graphs; in [28] Loebl and Sereni conjecture that the U-polynomial (equivalently the Tutte symmetric function XB) does.
It is natural to consider whether the chromatic symmetric function itself already distinguishes split graphs. Unfortunately it does not, and in particular, the following lemma allows for the construction of infinitely many pairs of split graphs that have equal chromatic symmetric functions. This construction will make use of graph automorphisms. An automorphism of a graph G is an isomorphism f of G with itself, and likewise a w-automorphism of a vertex-weighted graph (G, w) is a map f that is a w-isomorphism of (G, w) with itself. Additionally . For brevity if v ∈ V (G) we also use v to refer to the corresponding vertex of sp(G).
Lemma 8. Let G be an unweighted graph. Suppose G has (not necessarily distinct) vertices u, u ′ , v, v ′ such that: • There is some automorphism of G that maps u to u ′ , and some (possibly different) automorphism of G that maps v to v ′ .
Proof. Throughout this proof we will omit the weight function w from (G, w); the vertex weights will always all be 1 unless otherwise specified. Let G be as stated. In sp(G ∪ uv), let x be the splitting vertex of uv, and likewise in sp(G ∪ u ′ v ′ ) let x ′ be the splitting vertex of u ′ v ′ . By applying Lemma 7 to edge ux of sp(G ∪ uv) and edge u ′ x ′ of sp(G ∪ u ′ v ′ ) it suffices to show that the graphs sp(G ∪ uv)\ux and sp(G ∪ u ′ v ′ )\u ′ x ′ are w-isomorphic, and that the graphs sp(G ∪ uv) ∤ ux and sp( Note that if f : V (G) → V (G) is an automorphism of G, we may extend it to an automorphism of sp(G) by defining that for z ∈ V (sp(G))\V (G), if z is the splitting vertex of ab, f (z) is the splitting vertex of f (a)f (b). Let which may be extended to an automorphism of sp(G) as described above. It is easy to verify that if we extend f once more to a function f : We now address the graphs with contracted edges. Upon applying simple contraction to the edge ux ∈ sp(G ∪ uv), we let z be the vertex formed by contraction (now with weight 2), and as we are applying simple contraction. Likewise, when applying contraction to the edge u ′ x ′ ∈ sp(G ∪ u ′ v ′ ), we let z ′ be the vertex formed by contraction (now with weight 2). Let By hypothesis there is an automorphism of G taking u to u ′ that extends to an automorphism of sp(G). By extending f to a function f : Thus, when G∪uv is not isomorphic to G∪u ′ v ′ , the graphs sp(G∪uv) and sp(G∪u ′ v ′ ) are nonisomorphic split graphs with equal chromatic symmetric functions. One way to generate such examples easily is by taking an arbitrary noncomplete connected graph G, and choosing any nonedge ab in G. Then we construct 2G as the disjoint union of graphs G and G * , where G * is isomorphic to G (that is, V (2G) = V (G) ⊔ V (G * ), and E(2G) = E(G) ⊔ E(G * )). Let f : V (G) → V (G * ) be an isomorphism of G and G * . In the statement of Lemma 8, let u = u ′ = a and v = b be vertices of the component G, and v ′ = f (b) a vertex of the component G * . Then it is simple to verify that these choices for u, u ′ , v, v ′ satisfy the lemma, and that the two graphs 2G ∪ uv and 2G ∪ u ′ v ′ are nonisomorphic, since the latter is connected, and the former is not.
We can also use Lemma 8 to produce two nonisomorphic graphs, both connected, such that their split graphs have equal chromatic symmetric functions, as shown in Figure 2. Note that in this figure G ∪ uv is not isomorphic to G ∪ u ′ v ′ because, for example, G ∪ uv contains a triangle (K 3 ), and G ∪ u ′ v ′ does not. Figure 2: An unweighted connected graph G such that X sp(G∪uv) = X sp(G∪u ′ v ′ ) .
However, Lemma 8 can not generalize directly to XB because ∤ does not admit a simple deletion-contraction relation on XB. If we instead use normal contraction / on the edge ux, we get a multi-edge between u and v, and likewise for u ′ and v ′ . Thus, to generalize Lemma 8 we would need a single automorphism of G that takes u to u ′ and v to v ′ simultaneously; but then clearly G ∪ uv and G ∪ u ′ v ′ would be isomorphic!

Further Constructions of Graphs with Equal X G
In much of the recent literature on the chromatic symmetric function, examples of pairs of graphs with equal chromatic symmetric function have been generated using a result of Orellana and Scott. We reiterate it here and also prove that it extends to vertex-weighted graphs: Theorem 4.2). Let (G, w) be a simple, vertex-weighted graph with distinct vertices v 1 , v 2 , v 3 , v 4 such that Then the graphs G ∪ v 1 v 3 and G ∪ v 2 v 4 have equal chromatic symmetric function.
The portions of these graphs induced by v 1 , v 2 , v 3 , v 4 and their contractions are illustrated in Figure 3. It is clear from this figure that the automorphism f given by hypothesis induces a w-isomorphism of G 1 ∤ v 1 v 3 and G 2 ∤ v 2 v 4 , so we are done. 2 1, 3 4 1 2, 4 3 Figure 3: In addition to Lemma 8 and Theorem 9, we present one more method for constructing graphs with equal chromatic symmetric function. This method is inspired by the case u = u ′ of Lemma 8, but can be used in slightly more general contexts and is more akin to Theorem 9.
Given a simple graph G and a vertex v ∈ V (G), we define the neighborhood of v to be Lemma 10. Let (G, w) be a simple vertex-weighted graph, and let v 1 , v 2 , v 3 be distinct vertices of G satisfying • There is a w-automorphism f of G\v 3 such that f (v 1 ) = v 2 and f (v 2 ) = v 1 .
Then the graphs G ∪ v 1 v 3 and G ∪ v 2 v 3 have equal chromatic symmetric functions.
Proof. We let e 1 = v 1 v 3 and e 2 = v 2 v 3 be nonedges of G. By applying Lemma 7 to edge e 1 of G ∪ e 1 and to edge e 2 of G ∪ e 2 , it suffices to show that (G ∤ e 1 , w/e 1 ) and (G ∤ e 2 , w/e 2 ) are w-isomorphic (and from now on, we suppress mention of the weight functions).
In what follows, we let u 1 be the contracted vertex (of weight 2) in G ∤ e 1 , and we let u 2 be the contracted vertex (of weight 2) in G ∤ e 2 . Furthermore, G ∤ e 1 contains v 2 but not v 1 , and G ∤ e 2 contains v 1 but not v 2 , and otherwise these graphs have the same vertex set, all of weight 1 except for u 1 or u 2 .
Let f be the w-automorphism of G\v 3 that swaps v 1 and v 2 . We define the map g : V (G ∤ e 1 ) → V (G ∤ e 2 ) by g(v) = f (v) if v = u 1 , v 2 , g(u 1 ) = u 2 , and g(v 2 ) = v 1 . Clearly this g is a w-isomorphism if it is an isomorphism. All edge and nonedge relations between vertices of G ∤ e 1 other than u 1 and v 2 are preserved in G ∤ e 2 by g since they were preserved by f , so it suffices to look at edges and nonedges involving u 1 and v 2 .
Let G 1 = G ∤ e 1 and G 2 = G ∤ e 2 . Using the definition of contraction and the hypotheses we have so the neighborhood of u 1 is mapped to the neighborhood of u 2 by g. Additionally, so the neighborhood of v 2 is mapped to the neighborhood of v 1 , and this concludes the proof.

Graphs with Equal XB
Relative to X G , only a few examples of nonisomorphic graphs with equal Tutte symmetric functions are known. An example with a minimum number of vertices and edges is given by Markstrom [30] 3 . Additionally, Brylawski [7] uses the rotor-like graph given in Figure  4 to construct a family of graph pairs with arbitrarily high connectivity and equal Tutte symmetric function 4 . We modify Brylawski's result to construct graph pairs with arbitrarily high girth and equal Tutte symmetric function.
Theorem 11. Let k > 2 be a positive integer, and let (G, w) be a (not necessarily simple) vertex-weighted graph with distinct vertices a, b, and c such that there exists a wautomorphism f of (G, w) satisfying f (a) = b, f (b) = c, and f (c) = a. Modify the graph (G, w) by replacing every edge with an unweighted path of length k (that is, for each edge u 1 u 2 = e ∈ E(G), delete e, add weight-1 vertices e 1 , . . . , e k−1 to G, and add edges u 1 e 1 , e 1 e 2 , . . . , e k−1 u 2 to G). Then construct the vertex-weighted graphs (G 1 , w) and (G 2 , w) by retaining the weight function w and setting Then Proof. Note that the modified version of G still admits f as a w-automorphism with f (a) = b, f (b) = c, and f (c) = a by extending its definition to the newly formed vertices. We apply Lemma 7 to the edges bx 1 of (G 1 , w) and cx 1 of (G 2 , w). We may check that the graph (G 1 \bx 1 , w) is w-isomorphic to (G 2 \cx 1 , w) by extending f to a map f : we may verify that f is a w-isomorphism from (G 1 /bx 1 , w/bx 1 ) to (G 2 /cx 1 , w/cx 1 ) upon also letting f map the vertex formed by contraction in G 1 to the vertex formed by contraction in G 2 , and this concludes the proof.
Thus, we may use Theorem 11 to construct pairs of (unweighted) graphs of arbitrarily high girth with equal Tutte symmetric function given a single graph G such that the resulting graphs G 1 and G 2 are always nonisomorphic. It is straightforward to verify that the graph of Brylawski as given in Figure 4 is one such example. This construction is particularly noteworthy since there has not previously been shown even triangle-free graphs with the same chromatic symmetric function! a b c Figure 4: A graph G giving rise to nonisomorphic graphs G 1 and G 2 with the same Tutte symmetric functions and arbitrarily large girth.
We also give two pairs of small graphs with equal Tutte symmetric functions that do not fit the requirements of Theorem 11. In the figures that follow, the numbers next to the vertices are labels rather than weights, so that graphs can be redrawn to illustrate isomorphisms. Vertex weights from an edge contraction will be denoted by simply listing each original vertex label that corresponds to a vertex formed by edge contraction.
First, we consider the graphs shown in Figure 5. Let the graph on the left be called G 1 , and the graph on the right G 2 . First, note the graphs are indeed nonisomorphic, since for example G 1 has the vertex 1 with degree two that is not part of a triangle, but in G 2 both vertices of degree two are in triangles.
To show that these graphs have the same Tutte symmetric function, we apply Lemma 7 to the edge (6, 7) in both graphs, and reduce to showing that the edge-deleted graphs are w-isomorphic, and the edge-contracted graphs are w-isomorphic. It is easy to verify that both edge-contracted graphs are isomorphic to the graph in Figure 6. To see that the edge-deleted graphs are isomorphic, take the first graph, delete the edge (6, 7) and then rearrange the vertices as in Figure 7.  Using this figure, it is easy to see that G 1 \(6, 7) is w-isomorphic to G 2 \ (6, 7), and this shows that the graphs G 1 and G 2 have equal Tutte symmetric functions.
For a second example, we consider the graphs in Figure 8 (with the edges highlighted in red that we will apply Lemma 7). Let the graph on the top of this figure be H 1 , and the graph on the bottom be H 2 . First, we verify that H 1 and H 2 are nonisomorphic. Both graphs have exactly two vertices of degree 2, namely vertices 5 and 7 in H 1 and vertices 5 and 1 in H 2 . However, in H 2 these two vertices have a common neighbor, as they are both adjacent to 8, but the corresponding vertices do not have a common neighbor in H 1 .
To show that H 1 and H 2 nonetheless have equal Tutte symmetric function, we apply Lemma 7 to the edges marked in red in Figure 8. Clearly the graphs with these edges deleted are isomorphic. The contracted graphs are shown in Figure 9. The graphs in Figure 9 are not w-isomorphic, but it suffices to show that they have equal XB, which also provides an example of a pair of non-trivially weighted graphs with equal Tutte symmetric function. To do so, we again apply Lemma 7. First, in this figure, we rearrange the top graph H 1 /(1, 6) into the graph in Figure 10. Clearly now the graphs in Figure 9 are w-isomorphic with the red edges deleted. We show that they are also w-isomorphic with the red edges contracted. Those graphs correspond to H 1 and H 2 with two edges contracted and are shown in Figure 11. To illustrate that the two graphs in Figure 11 are w-isomorphic, we redraw the top graph as shown in Figure 12, making the w-isomorphism apparent.

Further Constructions via the W -polynomial
In [13], the second and third authors noted that vertex-weighted trees are not always distinguished by their chromatic symmetric functions, but the counterexample given in that paper had trees that were isomorphic when unweighted. Other examples of vertex-weighted paths with the same chromatic symmetric function appear in [28]. In this section we will develop an algebraic method to construct vertex-weighted graphs with the same W -polynomial; by Lemma 4 this method also yields vertex-weighted graphs with the same XB. The method extends some of the constructions in [4] by the first and fourth authors. In an independent work [2], Aliniaeifard, Wang and van Willigenburg obtained results similar to the ones presented in this section, but written completely in the language of symmetric functions. A 2-pointed vertex-weighted graph is a tuple (G, w, s, t), where (G, w) is a vertex-weighted graph, and s and t are (possibly the same) vertices of G. When w, s, and t are clear, we will often just write G in a slight abuse of notation. Given two 2-pointed vertex-weighted graphs (G, w, s, t) and (H, w ′ , s ′ , t ′ ), define (G, w, s, t) · (H, w ′ , s ′ , t ′ ) = (G · H, w · w ′ , s, t ′ ), where G · H is the graph formed by taking the disjoint union of G and H and then adding an edge between t and s ′ , and the weight function w · w ′ assigns to each vertex the weight it had in G or H. Second, define (G, w, s, t) ⊙ (H, w ′ , s ′ , t ′ ) as the 2-pointed vertex-weighted graph (G ⊙ H, w ⊙ w ′ , s, t ′ ) where G ⊙ H = (G · H)/e where e is the edge connecting t and s ′ , and w ⊙ w ′ = (w · w ′ )/e. Finally, if (G, w, s, t) is a 2-pointed vertex-weighted graph, its reversal is the 2-pointed vertex-weighted graph (G, w, t, s). When simply using G as shorthand to denote (G, w, s, t), we will write G * to denote its reversal.
Lemma 12. The following statements hold: 1. Given two 2-pointed vertex-weighted graphs (G, w, s, t) and (H, w ′ , s ′ , t ′ ), we have 2. For every composition β we have 3. Given two compositions β and β ′ , we have Proof. The first assertion follows directly from applying deletion-contraction to G · H and the edge e connecting t with s ′ and the definitions of · and ⊙. For the second assertion, we proceed by induction on the length of β. The base case follows easily, since if β = β 1 , then L β (x) = x β 1 , which is the same as the W -polynomial of an isolated vertex with weight β 1 . For the inductive step, we suppose that the assertion holds for compositions of length n and let β = β ′ β n+1 be any integer composition of length n + 1, where β ′ is a composition of length n. It is easy to check that Hence, by the induction hypothesis, we get from the last equality where the last equality follows from (18). This finishes the proof of the second assertion. The third assertion is a direct consequence of the previous two assertions.
Next, we will see how to use this lemma to construct arbitrarily large families of vertexweighted trees with the same W -polynomial by gluing together several copies of a seed graph. For a 2-pointed vertex-weighted graph G, we denote the graph G · G · . . . · G by G i and the graph G ⊙ G ⊙ . . . ⊙ G by G ⊙i (in each definition the graph G is repeated i times).
If G a 2-pointed vertex-weighted graph and β is an integer composition, then define Similarly, if γ is another integer composition, we can define It is not difficult to check that β • P γ = P β•γ and that where α, β are compositions, γ can be either a 2-pointed vertex-weighted graph or a composition, and * may be either · or ⊙.
Lemma 13. The operation • in the set of integer compositions is associative. Moreover, if (G, w, s, t) is a 2-pointed vertex-weighted graph and α and β are integer compositions, then Proof. The associativity of • for integer compositions is shown in [6,Proposition 3.3]. The second assertion will be shown by induction on the length of α. If α = α 1 , then which shows the base case. For the inductive step, given any composition α of length l, we may write it as α = α ′ α l where α ′ is a composition of length l − 1. By applying (20), we get By applying the induction hypothesis twice and then (20) again, the expresion in the right hand side of the last equation becomes This finishes the induction, and hence the proof.
The utility of the • operation is that the W -polynomial of β • G can be computed in terms of the L-polynomial of β and the W -polynomial of G.
Theorem 14. Let (G, w, s, t) be a 2-pointed vertex-weighted graph and β a composition. We have Proof. We do the proof by induction on the length l of β. For the base case, assume l = 1 and check that β • G = G ⊙β 1 and L β (x) = x β 1 . Thus, substituting x β 1 → W G ⊙β 1 (x, y) yields the assertion when l = 1. Now, using the inductive hypothesis, we will assume that the assertion holds for all compositions of length k and consider β = β ′ β k+1 , where β ′ = β 1 . . . β k . By the definition of thus, applying (18) yields It is easy to check that On the other hand, by the induction hypothesis, and , y)).
Combining these into (22), we get and thence the assertion follows by applying (19) to the last equation.
Corollary 15. Let β be an integer composition and (G, w, s, t) be a 2-pointed vertex-weighted graph. Then for every composition γ such that L γ = L β we have This motivates us to characterize the class of compositions with the same L-polynomial as a given composition. This characterization was obtained in [6] in the language of ribbon Schur functions and later recast in [4] to the language we are using here. If a composition β is written in the form β 1 • β 2 • · · · β k , then we call this a factorization of β. We say that the factorization β = β 1 • β 2 is trivial if any of the following conditions hold: 1. one of β 1 , β 2 are equal to 1, 2. the lengths of β 1 and β 2 are both equal to 1, 3. the compositions β 1 and β 2 both have all parts equal to 1. Remark 17. By the second assertion of Lemma 4 and (12) and the fact that X can be recovered from XB by setting t = −1 (which is equivalent to setting y = 0 in the Wpolynomial) we check that substituting each variable x i by −p i in L β yields the weighted chromatic symmetric function of P β . On the other hand, if we substitute each variable x i by −h i , where h i is the i-th complete homogeneous symmetric function, we obtain the ribbon Schur function associated with β. It follows that the homomorphism U of Λ defined by sending each p λ to h λ and then extending linearly transforms the chromatic symmetric function of the weighted path P β into the the ribbon Schur function associated with β. This observation is implicit in [4] and the morphism U is studied in detail in [2].
Combining Corollary 15 and Theorem 16 we get Corollary 18. Let (G, w, s, t) be a 2-pointed weighted graph and β an integer composition. Suppose we have an irreducible factorization β = β 1 • β 2 • · · · • β k . Then, all vertex-weighted graphs in the set have the same W -polynomial as β • G.  Finally, we apply these results to give two examples of weighted trees with the same W -polynomial (and hence Tutte symmetric function). For the first example we consider the 2-pointed vertex-weighted tree T depicted in Figure 13 and the composition β = 12. Then, it follows that the weighted trees β • T and β * • T have the same W -polynomial but they are not w-isomorphic. In fact, they are not even isomorphic as unweighted trees. These trees are also shown in Figure 13. However, in this case, there is an alternate way of checking that the trees have the same W -polynomial: It may be checked that the graphs are w-isomorphic when adding the dashed non-edge and when contracting the dashed non-edge, so it follows from Lemma 7. Our construction can also be used to obtain a more complicated pair of examples (that cannot be constructed by the direct use of deletion-contraction operations). Consider β • T and γ • T where T is the same 2-pointed vertex-weighted tree depicted in Figure 13, β = 12132 = 12•12 and γ = 13212 = 21•12. By applying Corollary 15 we see that these weighted trees have the same W -polynomial, but clearly they are not w-isomorphic; they are depicted in Figure 14.

Further Research
We conclude with some data and further possible considerations for research.
Using deletion-contraction relations, we computed X G and XB G for simple graphs with at most 8 vertices using data provided by [32]. We found many pairs of such graphs with equal chromatic symmetric function, and for all of these pairs we also determined whether the graphs are distinguished by XB. This information and more may be viewed at [11].
We find that triangles seem to play an important role in graphs with equal chromatic symmetric function. Note that in the 1000 pairs of graphs with equal chromatic symmetric function noted in [11], every graph contains a triangle. Furthermore, each of the three methods given in Sections 5.1 and 5.2 for constructing graphs with equal chromatic symmetric function always produces a pair of graphs containing triangles. In the case of Lemma 8 and Theorem 9 this is explicit. In the case of Lemma 10, suppose that we have a graph G satisfying the assumptions of the lemma. If N(v 3 ) = ∅, then G ∪ v 1 v 3 is isomorphic to G ∪ v 2 v 3 since by assumption there is an automorphism of G\v 3 swapping v 1 and v 2 . If there is a vertex x ∈ N(v 3 ), then by assumption also x ∈ N(v 1 ) and x ∈ N(v 2 ), so in G ∪ v 1 v 3 there is a triangle with vertices v 1 , v 3 , x and in G ∪ v 2 v 3 there is a triangle with vertices v 2 , v 3 , x. Thus, every G satisfying the conditions of Lemma 10 either produces two isomorphic graphs, or two graphs with equal chromatic symmetric function that both contain triangles. Finally, we also note the recent result of Penaguiao [39] showing that given any two nonisomorphic graphs with equal chromatic symmetric function, one may be transformed into the other by a finite number of applications of the triangular modular relation of Orellana and Scott [36].
Indeed, prior to the discovery of the construction given in Section 6.3, the authors considered whether triangle-free graphs may be distinguished by the chromatic symmetric function! It would be useful if it could be demonstrated an explicit sequence of applications of the modular relation of [36] that takes these triangle-free graphs to each other. As far as distinguishing graphs goes, the next logical question is to determine whether there are bipartite graphs with equal chromatic symmetric function.
The spanning tree formula (17) for the chromatic symmetric function is new, and may be useful to ongoing research. Furthermore, the sum runs over those spanning trees of a graph G with no external activity. It is worth noting that the number of this particular kind of spanning tree in a graph G is equal to the number of G-parking functions with respect to any vertex, and the number of acyclic orientations of G with exactly one sink [5]. It would be interesting to see if there are similar expansions to (17) that run over one of these sets.
Additionally, there is an expansion of the Tutte polynomial as a sum over G-parking functions given in [8] as where cb(f ) is the number of critical bridge vertices of G with respect to f (as defined in [8]), and w(f ) = |E(G)| − |V (G)| − v∈V (G) f (v). In the same way that the spanning tree formula (15) for XB is an extension of a similar formula for T G , perhaps there is a natural formula for XB that extends this G-parking function expansion of T G . Finally, there appear to be many rich, unexplored connections between the Tutte symmetric function and other functions derived from the V -polynomial. For example, one can use a specialization of the V -polynomial as a natural list-coloring polynomial [20]. It would be interesting to see if the Tutte symmetric function could be modified to consider this or other V -polynomial specializations.