On the Signless Laplacian Spectral Radius of C4-free k-cyclic Graphs

A k-cyclic graph is a connected graph of order n and size n + k-1. In this paper, we determine the maximal signless Laplacian spectral radius and the corresponding extremal graph among all C_4-free k-cyclic graphs of order n. Furthermore, we determine the first three unicycles and bicyclic, C_4-free graphs whose spectral radius of the signless Laplacian is maximal. Similar results are obtained for the(combinatorial)

Largest signless Laplacian spectral radius of uniform supertrees with diameter and pendent edges(vertices)

Let S(m,d,k)be the set of k-uniform supertrees with m edges and diameter d,and S1(m,d,k)be the k-uniform supertree obtained from a loose path u_(1),e_(1),u_(2),e_(2),...,u_(d),e_(d),u_(d+1),with length d by attaching m-d edges at vertex u_[d/2]+1.In this paper,we mainly determine S1(m,d,k)with the largest signless Laplacian spectral radius in S(m,d,k)for 3≤d≤m-1.We also determine the supertree with the second largest signless Laplacian spectral radius in S(m,3,k).Furthermore,we determine the unique/c-uniform supertree with the largest signless Laplacian spectral radius among all fc-uniform supertrees with n vertices and pendent edges(vertices).

On the Second Smallest and the Largest Normalized Laplacian Eigenvalues of a Graph

Let G be a simple connected graph with order n.Let L(G)and Q(G)be the normalized Laplacian and normalized signless Laplacian matrices of G,respectively.Letλk(G)be the k-th smallest normalized Laplacian eigenvalue of G.Denote byρ(A)the spectral radius of the matrix A.In this paper,we study the behaviors ofλ2(G)andρ(L(G))when the graph is perturbed by three operations.We also study the properties ofρ(L(G))and X for the connected bipartite graphs,where X is a unit eigenvector of L(G)corresponding toρ(L(G)).Meanwhile we characterize all the simple connected graphs withρ(L(G))=ρ(Q(G)).