Spectral radius conditions for the rigidity of graphs

Rigidity is the property of a structure that does not ﬂex under an applied force. In the past several decades, the rigidity of graphs has been widely studied in discrete geometry and combinatorics. Laman (1970) obtained a combinatorial characterization of rigid graphs in R 2 . Lov´asz and Yemini (1982) proved that every 6-connected graph is rigid in R 2 . Jackson and Jord´an (2005) strengthened this result, and showed that every 6-connected graph is globally rigid in R 2 . Thus every graph with algebraic connectivity greater than 5 is globally rigid in R 2 . In 2021, Cioab˘a, Dewar

we consider the following problem: Problem 1. Which spectral conditions can guarantee that a graph is rigid or globally rigid in R 2 ?
For a graph G, let D(G) denote the diagonal matrix of vertex degrees of G, and A(G) denote the adjacency matrix of G. The Laplacian matrix of G is defined as L(G) = D(G) − A(G). The second least eigenvalue of L(G), denoted by µ(G), is known as the algebraic connectivity of G. As the vertex-connectivity of G is not less than µ(G), the results in [21,15] imply that if µ(G) > 5 then G is globally rigid in R 2 . Based on some necessary conditions for packing rigid subgraphs, Cioabȃ, Dewar and Gu [3] strengthened this result, and proved that a graph G with minimum degree δ ! 6 is rigid in R 2 if µ(G) > 2 + 1 δ−1 , and is globally rigid in R 2 if µ(G) > 2 + 2 δ−1 . In this paper, we focus on giving some answers to Problem 1 in terms of the (adjacency) spectral radius of graphs. The spectral radius of a graph G, denoted by ρ(G), is the largest eigenvalue of its adjacency matrix A(G). A graph is k-connected if removing fewer than k vertices always leaves the remaining graph connected. Let K n denote the complete graph on n vertices, and B i n,n 1 denote the graph obtained from K n 1 ∪ K n−n 1 by adding i independent edges (with no common endvertex) between K n 1 and K n−n 1 . The main results are as follows.
Hendrickson [13] proved that every globally rigid graph in R d with at least d + 2 vertices is (d + 1)-connected and redundantly rigid. Thus it is necessary to assume that G is 3-connected when we consider the global rigidity of G in R 2 .
A graph G is minimally rigid if G is rigid but G − e is not rigid for all e ∈ E(G). Note that a graph is rigid if and only if it has a minimally rigid spanning subgraph. In 1970, Leman [20] provided a characterization for minimally rigid graphs in R 2 by using the edge count property, and proved that a graph G with n vertices and m edges is a minimally rigid if and only if m = 2n − 3 and e G (X) " 2|X| − 3 for all X ⊆ V (G) with |X| ! 2, where e G (X) is the number of edges of the subgraph G[X] induced by X in G. Minimally rigid graphs are also called Leman graphs in R 2 .
The join of two graphs G and H, denoted by G∇H, is the graph obtained from G ∪ H by adding all possible edges between G and H. Based on Leman's characterization for minimally rigid graphs in R 2 , we determine the unique graph attaining the maximum spectral radius among all connected minimally rigid graphs of order n in R 2 .

Preliminaries
In this section, we list some basic concepts and lemmas which will be used later.
Let M be a real n × n matrix, and let X = {1, 2, . . . , n}. Given a partition π : X = X 1 ∪ X 2 ∪ · · · ∪ X k , the matrix M can be correspondingly partitioned as The quotient matrix of M with respect to π is defined as the k × k matrix where b i,j is the average value of all row sums of M i,j . The partition π is equitable if each block M i,j of M has constant row sum b i,j . In this situation, the corresponding quotient matrix B π is also called equitable. Recall that B i n,n 1 denotes the graph obtained from K n 1 ∪K n−n 1 by adding i independent edges between K n 1 and K n−n 1 . Proof. Since B i n,a contains K n−a as a proper subgraph, we have ρ(B i n,a ) > ρ(K n−a ) = n − a − 1. Note that A(B i n,a ) has the equitable quotient matrix By a simple calculation, the characteristic polynomial of C a π is Also note that A(B i n,a+1 ) has the equitable quotient matrix C a+1 π , which is obtained by replacing a with a + 1 in C a π . As n ! 2a + 2, we have for all x ! n − a − 1. This implies that λ 1 (C a+1 π ) < λ 1 (C a π ). Therefore, by Lemma 5, we have ρ(B i n,a+1 ) < ρ(B i n,a ), and the result follows.
Lemma 7. (See [14,22]) Let G be a graph on n vertices and m edges with minimum degree δ ! 1. Then with equality if and only if G is either a δ-regular graph or a bidegreed graph in which each vertex is of degree either δ or n − 1.
Lemma 8. (See [14,22]) For nonnegative integers p and q with 2q " p(p − 1) and Lemma 9. Let a and b be two positive integers. If a ! b, then Thus the result follows.
be the subgraph of G induced by X, and let e G (X) be the number of edges in G[X]. Particularly, let e(G) = e G (V (G)) denote the number of edges of G. For X, Y ⊆ V (G), we denote by E G (X, Y ) the set of edges with one endpoint in X and one endpoint in Y , and e G (X, Y ) = |E G (X, Y )|. In particular, let Lemma 10. (See [12]) Let G be a graph with minimum degree δ and U be a non-empty For any partition π of V (G), let E G (π) denote the set of edges in G whose endpoints lie in different parts of π, and let e G (π) = |E G (π)|. A part is trivial if it contains a single vertex. Let Z ⊂ V (G), and let π be a partition of The following three lemmas about rigid graphs will play crucial roles in the proof of our main theorems.
Lemma 11. (See [11]) A graph G contains k edge-disjoint spanning rigid subgraphs if for every Z ⊂ V (G) and every partition π of V (G − Z) with n 0 trivial parts and n ′ 0 nontrivial parts, Lemma 12. (See [4,16]) Let G be a graph. Then G is globally rigid if and only if either G is a complete graph on at most three vertices or G is 3-connected and redundantly rigid.

Proof of the main theorems
In this section, we shall give the proofs of Theorems 2-4.
Proof of Theorem 2. Assume to the contrary that G is not rigid. Then G contains no spanning rigid subgraphs. By Lemma 11, there exist a subset Z of V (G) and a partition and therefore, We have the following two claims.
Again by Lemma 9, we obtain e(G) " max Combining this with (4), we have δ < 4, which is also impossible.
This completes the proof.
Recall that, for any partition π of V (G), E G (π) denotes the set of edges in G whose ends lie in different parts of π, and e G (π) = |E G (π)|.
Proof of Theorem 3. Assume to the contrary that G is not globally rigid. Since G is a 3-connected graph with minimum degree δ ! 6 and order n ! 2δ + 4, by Lemma 12, we see that G is not redundantly rigid. This suggests that there exists an edge f of G such that G − f is not rigid. Furthermore, by Lemma 11, there exist a subset Z of V (G) and a partition π of V (G − f − Z) with n 0 trivial parts v 1 , v 2 , . . . , v n 0 and n ′ 0 nontrivial parts First we assume that f ∈ E G−Z (π). Then e G−f −Z (π) = e G−Z (π) − 1. By (8), Recall that n Z (π) = and hence We have the following two claims.
In this case, the proof is similar as in Case 2 of Theorem 2, and we omit it. Now we assume that f / ∈ E G−Z (π). Then By similar arguments as above, we also can deduce a contradiction. This completes the proof.
Proof of Theorem 4. Suppose that G has the maximum spectral radius among all minimally rigid graphs of order n ! 3. By Lemma 13, we have e(G) = 2n − 3 and e G (X) " 2|X| − 3 for all X ⊆ V (G) with |X| ! 2. Note that K 2 ∇(n − 2)K 1 is a minimally rigid graph. Then Let δ denote the minimum degree of G. We assert that δ ! 2. In fact, if there exists some vertex u ∈ V (G) such that d Thus the equalities hold in (16) and (17). It follows that δ = 2 and G is either a 2-regular graph, or a bidegreed graph in which each vertex is of degree 2 or n − 1 by Lemma 7. If n = 3, then G ∼ = K 3 , as required. Now suppose that n ! 4. Let t = |{v ∈ V (G)| d G (v) = n−1}|. If 0 " t " 1, then e(G) < 2n − 3, and if t ! 3 then e(G) > 2n − 3, both are impossible. Thus t = 2, and G ∼ = K 2 ∇(n − 2)K 1 . This completes the proof.

Concluding remarks
In this paper, we provide a spectral radius condition for the rigidity (resp., globally rigidity) of 2-connected (resp., 3-connected) graphs with given minimum degree in R 2 . In particular, we give the answers to Problem 1 for k = 2, 3. Note that every 6-connected graph is rigid (resp., globally rigid). Thus, the Problem 1 becomes more involved for k = 4, 5. When k = 4, 5, by using similar analysis as Theorems 2 and 3, we can obtain that a k-connected graph G is rigid (resp., globally rigid) if ρ(G) > ρ(B k n,δ+1 ). As B k n,δ+1 is both rigid and globally rigid for k = 4, 5, we end the paper by proposing the following problem for further research.