Suppose π = (d1, d2, . . . , dn) and π = (d1, d ′ 2 , . . . , dn) are two positive nonincreasing degree sequences, write π ⊳ π if and only if π 6= π, ∑n i=1 di = ∑n i=1 d ′ i, and ∑j i=1 di ≤ ∑j i=1 di for all j = 1, 2, . . . , n. Let ρ(G) and μ(G) be the spectral radius and signless Laplacian spectral radius of G, respectively. Also let G and G be the extremal graphs with the maximal (signless...

Suppose that the vertex set of a connected graph G is $$V(G)=\{v_1,\ldots ,v_n\}$$ . Then we denote by $$Tr_{G}(v_i)$$ sum distances between $$v_i$$ and all other vertices G. Let Tr(G) be $$n\times n$$ diagonal matrix with its (i, i)-entry equal to $$Tr_{G}(v_{i})$$ D(G) distance $$Q_{D}(G)=Tr(G)+D(G)$$ signless Laplacian The largest eigenvalues $$Q_D(G)$$ called spectral radius In this ...

For a simple graph G, let e(G) denote the number of edges and Sk(G) denote the sum of the k largest eigenvalues of the signless Laplacian matrix of G. We conjecture that for any graph G with n vertices, Sk(G) ≤ e(G) + k+1 2 for k = 1, . . . , n. We prove the conjecture for k = 2 for any graph, and for all k for regular graphs. The conjecture is an analogous to a conjecture by A.E. Brouwer with ...

SLEE has various applications in a large variety of problems. The signless Laplacian Estrada index hypergraph H is defined as SLEE(H)=∑i=1neλi(Q), where λ1(Q),λ2(Q),…,λn(Q) are the eigenvalues matrix H. In this paper, we characterize unique r-uniform unicyclic hypergraphs with maximum and minimum SLEE.

Let q (G) denote the spectral radius of the signless Laplacian matrix of a graph G, also known as the Q-index of G. The aim of this note is to study a general extremal problem: How large can q (G) be when G belongs to an abstract graph property? Even knowing very little about the graph property, this paper shows that useful conclusions about the asymptotics of q (G) can be made, which turn out ...

A graph G is said to be determined by its Q-spectrum if with respect to the signless Laplacian matrix Q , any graph having the same spectrum as G is isomorphic to G. The lollipop graph, denoted by Hn,p, is obtained by appending a cycle Cp to a pendant vertex of a path Pn−p. In this paper, it is proved that all lollipop graphs are determined by their Q -spectra. © 2008 Elsevier B.V. All rights r...

LetG be a finite, simple, and undirected graphwith n vertices. Thematrix L(G) = D(G)−A(G) (resp., L+(G) = D(G)+A(G)) is called the Laplacianmatrix (resp., signless Laplacianmatrix [1–4]) of G, where A(G) is the adjacency matrix and D(G) is the diagonal matrix of the vertex degrees. (For details on Laplacian matrix, see [5, 6].) Since A(G), L(G) and L+(G) are all real symmetric matrices, their e...

We determine the graph with the largest signless Laplacian spectral radius among all unicyclic graphs with fixed matching number.

This paper gives a tight upper bound on the spectral radius of the signless Laplacian of graphs of given order and clique number. More precisely, let G be a graph of order n, let A be its adjacency matrix, and let D be the diagonal matrix of the row-sums of A. If G has clique number ω, then the largest eigenvalue q (G) of the matrix Q = A+D satisfies q (G) ≤ 2 (1− 1/ω)n. If G is a complete regu...