The Laplacian spread of Cactuses

Connected graphs in which any two of its cycles have at most one common vertex are called cactuses. In this paper, we continue the work on Laplacian spread of graphs, and determine the graph with maximal Laplacian spread in all cactuses with n vertices.


Introduction
Let G = (V, E) be a simple connected graph with vertex set denotes the diagonal matrix of vertex degrees of G.It is easy to see that L(G) is a positive semidefinite symmetric matrix and its rows sum to 0, so L(G) is singular.Denote its eigenvalues by µ 1 (G) ≥ µ 2 (G) ≥ • • • ≥ µ n (G) = 0, which are always enumerated in non-increasing order and repeated according to their multiplicities.We call the largest eigenvalue of L(G) the Laplacian spectral radius of the graph G, denoted by µ(G).Moreover, since G is connected if and only if µ n−1 (G) > 0, Fielder [2,3] thought, in a sense, µ n−1 (G) as a quantitative measure of connectivity.Hence he called µ n−1 (G) the algebraic connectivity of G and denoted by α(G).The Laplacian spread of the graph G is defined to be The notion of Laplacian spread is very significant in characterizing the global structural property of graphs.Recently, Fan et al [4] have showed that, among all trees with n vertices, K 1,n−1 and P n are the unique trees with the maximal Laplacian spread and minimal Laplacian spread, respectively.Bao [1] et al determined the graph with maximal in all unicyclic graphs.
Connected graphs in which any two of its cycles have at most one common vertex are called cactuses.Throughout this paper, the characteristic polynomial det(xI row and column corresponding to the vertex v. Let C n be the set of cactuses with n vertices.In this paper, we continue the work on Laplacian spread of graphs, and determine the graph class C 1 (see Fig. 1) with maximal Laplacian spread in all C n (n ≥ 10), where C 1 is the graph class by gluing one vertex of s triangles to the maximum degree vertex of K 1,r (2s + r = n − 1), respectively.Now, we give the main theorem in this paper: 2 Proof of Theorem 1 The following lemmas are necessary for our main results.Denote by ) Let G be a connected graph of n vertices and with a cutpoint v. Then the equality holds if and only if v is adjacent to every other vertex of G.
Lemma 6 [11] Let G be a connected graph on n vertices.Suppose that Lemma 7 ( [6]) Let G = G 1 u : vG 2 be the graph obtained by joining the vertex u of the graph G 1 to the vertex v of the graph G 2 by an edge.Then From (1), Lemma 3 and by Lemma 5, for

Combining (1) and Lemma 2, we know if
So in the followings we may always assume that there exist u, v ∈ V (G) such that We introduce some notations.Let T i (p, q) (i = 1, 2, 3) be a tree obtained from P i+1 by adding p, q pendent edges to a pendent vertex, respectively.
Each unicyclic graph can be obtained by attaching rooted trees to the vertices of a cycle In order to prove our main theorem, we give three graph classes G 1 , G 2 and G 3 .Fig. 2:
Using the similar method, we can obtain the same result.
If G ∈ G 3 for some r, s, by Lemma 9, we have for n ≥ 5, µ(G) < n − 1. 2 From the above lemmas, we give the proof of Theorem 1.The proof is completed.2

Proof of
and edge set E. Denote by d i (or d(v i )) the degree of the vertex v i of the graph G. Let A(G) be the adjacency matrix of G and L(G) = D(G)−A(G) be the Laplacian matrix of the graph G, where D(G) =diag(d 1 , d 2 , • • • , d n )

Lemma 3 (
[5,9]) Let G be a connected graph on n vertices with at least one edge, then µ(G) ≥ ∆(G)+ 1, where ∆(G) is the maximum degree of the graph G, with equality if and only if ∆(G) = n − 1.Lemma 4 ([8]) Suppose u, v are two distinct vertices of a connected graph H. Let G be the graph obtained from H by attaching s new paths which contradicts the assumptions of |N (u) ∪ N (v)| = n, so we only discuss the case uv ∈ E(G).If uv ∈ E(G), then by Lemma 8, we have |N (u) ∩ N (v)| ≤ 1.