Alon-Boppana-type bounds for weighted graphs

The unraveled ball of radius $r$ centered at a vertex $v$ in a weighted graph $G$ is the ball of radius $r$ centered at $v$ in the universal cover of $G$. We present a general bound on the maximum spectral radius of unraveled balls of fixed radius in a weighted graph. The weighted degree of a vertex in a weighted graph is the sum of weights of edges incident to the vertex. A weighted graph is called regular if the weighted degrees of its vertices are the same. Using the result on unraveled balls, we prove a variation of the Alon-Boppana theorem for regular weighted graphs.


Introduction
In 1993, Freidmen [1] refined the celebrated Alon-Boppana theorem [5].He proved that for every d-regular graph G with diameter 2r, the second largest eigenvalue of adjacency matrix of G, denoted by λ 2 (G), satisfies In 2005, Hoory [2, Theorem 1] studied the spectral radius of the universal cover of a non-regular graph.As a corollary he proved a variation of the Alon-Boppana theorem for graphs with r-robust average degree at least d, which was later improved by Jiang [3]; see also [4,Lemma 19].We say that a graph has r-robust average degree at least d if the average degree of a graph is at least d after deleting any ball of radius r.We denote by P r the path with r vertices and spectral radius λ 1 (P r ) = 2 cos( π r+1 ).To prove Theorem 1, Jiang studied the maximum spectral radius of unraveled balls of a graph G, which are balls in the universal cover of G; see Definition 1.In 2022, Wang and Zhang applied the machinery developed in [3] and proved an analog of Theorem 1 for the normalized Laplacian of a graph [7,Theorem 1.6], improving the bounds from [8]; see also [6].To show this result, they studied the maximum spectral radius of unraveled balls of a weighted graph.
Motivated by the works of Jiang and Wang-Zhang, we develop further their ideas and prove a generalization of Wang and Zhang's result on the spectral radius of unraveled balls for a weighted graph; see Theorem 4. This implies an analog of Theorem 1 for regular weighted graphs, as shown in Theorem 3. A weighted graph is a graph without parallel edges and loops, in which every edge is assigned to a positive number.Formally, a weighted graph G is a triple (V (G), E(G), w G ), where V (G) and E(G) are the vertex and edge sets of the graph G, respectively, and w G : E(G) → R + is the weight function, with R + being the set of positive real numbers.For sake of brevity, we write w ab and w ba for the weight w G (ab) of an edge ab ∈ E(G).The weighted degree of a vertex v, denoted by w v , is the sum of the weights of the edges incident to v, that is, w v = vu∈E(G) w vu .A weighted graph is called w-regular if the weighted degree of every vertex equals w.Throughout the paper, we regularly write "a weighted graph with minimal degree at least 2 ", which means that each vertex is incident to at least 2 edges (rather than the weighted degree of each vertex is at least 2).
A non-backtracking walk of length n in a weighted graph is a sequence of vertices (v 0 , . . ., v n ) such that any two consecutive are adjacent and v i = v i+2 for all i ∈ {0, . . ., n− 2}.Denote by W i (G) the set of non-backtracking walks on a graph G of length i.
Definition 1.Given a weighted graph G, we define the weighted tree G(v, r) as follows.Its vertex set is the set of all non-backtracking walks of length at most r that start at v, where two vertices are adjacent if one is a simple extension of the other.Specifically, vertices (v 0 , . . ., v n ) and (u 0 , . . ., u m ) with n < m are adjacent if and only if m = n + 1 and v i = u i for all i ∈ {0, . . ., n}.We say this edge of G(v, r) is extended by the edge u m−1 u m in the graph G. Two vertices of the same length are never adjacent.We assign a weight to each edge in G(v, r) equal to the weight of its extending edge in G.
In other words, the graph G(v, r), which we call an unraveled ball, is isomorphic to a ball of radius r in the universal cover G of G. Slightly abusing notation, in the current paper, we say G(v, r) is an induced subgraph of G It is worth mentioning that we may look at the set W 1 (G) as the set of directed edges of a graph G, that is, for any edge xy ∈ E(G), there are two corresponding non-backtracking edges (x, y) and (y, x) in W 1 (G).So, we write w (x,y) for w xy if xy ∈ E(G).
The weighted adjacency matrix of a weighted graph G is denoted by A G .Let λ i (G) be the i-th eigenvalue of G.The main corollary of Theorem 4, the central result of the paper, is presented below.
Theorem 2. Let G be a w-regular weighted graph with minimum degree at least 2. Then for any positive integer r, there is a vertex v such that We define the average combinatorial degree of a weighted graph G as 2|E(G)| |V (G)| .Using Theorem 5, we obtain a variation of the Alon-Boppana theorem.
Theorem 3. Let G be a w-regular weighted graph with combinatorial degree d ≥ 39 and let r be any positive integer.Assume that for every vertex v there is an induced subgraph of G \ G(v, r + 1) with minimum weighted degree at least 2w A slightly more involved argument allows to show that Theorem 3 holds for d ≥ 7.1980 . . .; see Remark 2.
The rest of the paper is organized as follows.Section 2 presents the proof of the main result of the paper, Theorem 4, which requires auxiliary notation.In Section 3, we derive Corollary 5, slightly generalizing Theorem 2, as well as its slightly weaker form, Corollary 6.Finally, we establish Theorem 3 in Section 4.
Acknowledgments.We thank Alexander Golovanov for fruitful discussions and careful proofreading of the paper.
2. Bounding the maximum spectral radius of an unraveled ball Definition 2. Given a weighted graph G with minimum degree at least 2, a stationary Markov chain We say that e 2 ∈ W 1 (G) prolongs e 1 ∈ W 1 (G) if there are three distinct vertices x, y, z ∈ V (G) such that e 1 = (z, y) and e 2 = (y, x).For the sake of brevity, we write e 1 → e 2 if e 2 prolongs e 1 .
If the minimum degree of a weighted (connected) graph G is at least 2, then a stationary Markov chain assigned to G has no absorbing states and its stationary distribution π = (π e ) e∈W 1 (G) is well defined.Since the Markov chain is stationary, we have Pr E i = e = π e for any positive integer i.
Since the ending vertex of E i and the starting vertex of E i+1 are the same, we can concatenate E 1 , . . ., E i to form a random non-backtracking walk of length i, denoted by the random variables Y i = (X 0 , . . ., X i ).
With these definitions, we can state and prove the following general theorem.
Theorem 4. Let G be a weighted graph with minimum degree at least 2, let (E i ) +∞ i=1 be a stationary Markov chain assigned to G with transition matrix P = (p e 1 ,e 2 ) e 1 ,e 2 ∈W 1 (G) and stationary distribution π = (π e ) e∈W 1 (G) .For any function g : W 1 (G) → R and a positive integer r, there is a vertex v of G such that .
Proof.Let us begin by defining the weighted forest F G as the union of all graphs G(v, r), .
Denote by (x 1 , . . ., x r+1 ) ∈ R r+1 the eigenvector of the spectral radius λ 1 (P r+1 ) of the path P r+1 of length r.Then the Rayleigh principle yields Define a vector f ∈ R V (F ) by setting, for ω Therefore, we have Denoting ω − = (v 0 , . . ., v i−1 ) and For the Markov chain, we have Substituting this in the equation for f, A F G f , we get Since Pr .

Proofs of corollaries on regular weighted graphs
To prove special cases of Theorem 4, the authors of [3, Thereom 1] and [7, Theorem 1.5] consider the following stationary Markov chain on W 1 (G) such that its stationary distribution can be easily found.Namely, they assumed that given the stage One can easily verify that the distribution π = (π e ) e∈W 1 (G) with π e = 1 |W 1 (G)| is stationary.In the next general corollary of Theorem 4, we use another stationary Markov chain assigned to a regular weighted graph such that its stationary distribution be explicitly found.
Corollary 5. Let G be a w-regular weighted graph with minimum degree at least 2. For any function g : W 1 (G) → R and any positive integer r, there is a vertex v of G such that .
Applying Theorem 4 to this Markov chain, we easily obtain the first desired inequality.Assuming that g(e) = (w − w e ) −1/2 , we have The multipliers of the right-hand side of (3) can be easily found Substituting these equalities in (3), we obtain the second desired inequality.
Corollary 6.Let G be a w-regular weighted graph with minimum degree at least 2.
Assume that its average combinatorial degree (that is, d ≥ 38.7620 . . .).Then, for any positive integer r, we have Proof.There are two possible cases.Case 1.
There is e = (v, u) ∈ W 1 (G) such that w e ≥ µw.Define a vector f ′ ∈ R V ( G(v,r)) by setting Using the Rayleigh principle, we have which finishes the proof in this case as 2 √ d − 1/d ≤ µ and λ 1 (P r+1 ) < 2.
Case 2. For any vertex e ∈ W 1 (G), we have w e ≤ µw.By a straightforward computation, we obtain that the function x → x 3/2 (w − x) 1/2 is convex on the interval [0, µw] and concave on [µw, w], where µ = 3− √ 3 4 .By Jensen's inequality, we obtain Substituting this inequality in the second inequality of Corollary 5, we finish the proof.
Remark 2. Slightly modifying the argument, we can prove Corollary 6 for d ≥ 1/t 0 = 7.1980 . . ., where t 0 is defined below.Let g : [0, w] → R be a function given by g(y) = y 3/2 (w − y) 1/2 .For any t ∈ (0, 1), let ℓ t : R → R be an affine function defining the tangent line of g at the point tw ∈ (0, w); see Figure 1.Consider the following system of equations and inequalities which has only one solution: x = x 0 = 0.6917 . . ., t = t 0 = 0.1389 . . .; see Figure 1.First, we may assume that for all e ∈ W 1 (G); otherwise, we use the proof of the first case in Corollary 6.
Next, consider the function h : [0, x 0 w] → R defined by and ℓ t 0 defines the tangent line for the graph of g at the point t 0 w, we conclude that the function h is convex on [0, x 0 w] and g(y) ≥ h(y) for any 0 ≤ y ≤ x 0 w.Therefore, we can apply Jensen's inequality for h and obtain the desired inequality (Here we use that d ≥ 1/t 0 , and thus h( w d ) = g( w d ) by the definition of h.) Substituting this inequality in the second inequality of Corollary 5, we finish the proof.Remark 3. Using Corollaries 5 and 6, we can prove lower bounds for the spectral radius for the universal cover G of a w-regular weighted graph G as it is shown in Remark 1.

Proof of Theorem 3
The following lemma connects the spectral radii of balls G(v, r) and G(v, r).Suppose that G has a vertex of degree 1, that is, incident to one edge.Then there is a connected component of G that is a path of length 1 with weight of its only edge equal to w.Clearly, this component has eigenvalue w, which is large enough to finish the proof in this case.Therefore, we can assume that there are no vertices of degree 1 in G.
Lemma 7 and Corollary 6 yield that there exists a vertex v ∈ G such that Denote by f 1 ∈ R V (G) the vector that coincides on V (G(v, r)) with the eigenvector of the spectral radius of G(v, r) and is zero on V (G) \ V (G(v, r)).By the Rayleigh principle, we have Let G ′ be an induced subgraph of G \ G(v, r + 1) with minimum weighted degree at least 2w √ d − 1/d.Define a vector f 2 ∈ R V (G) by setting Hence by the Rayleigh principle, we obtain One can choose scalars c 1 and c 2 such that the vector f = c 1 f 1 + c 2 f 2 = 0 is perpendicular to the eigenvector (1, . . ., 1) of the spectral radius λ 1 (G) = w.Therefore, by the Rayleigh principle, we obtain which finishes the proof.

Theorem 1 .
[3, Theorem 8]  Let d ≥ 1 be a real number and let r be a positive integer.If a graph G has an r-robust average degree at least d, thenλ 2 (G) λ 1 (P r ) ≥ √ d − 1.
uniformly at random.Hence the transition matrix P = (p e 1 ,e 2 ) e 1 ,e 2 ∈W 1 (G) of this Markov chain is defined by p (x,y),(z,t) =    1 deg y − 1 if y = z and t = x; 0 otherwise.
2010 Mathematics Subject Classification.05C22.The research of A.P. was supported by the Ministry of Education and Science of Bulgaria, Scientific Programme "Enhancing the Research Capacity in Mathematical Sciences (PIKOM)", No. DO1-67/05.05.2022.A.P. is a Young Russian Mathematics award winner and would like to thank its sponsors and jury.