Let B be a weighted generalized Bethe tree of k levels (k > 1) in which nj is the number of vertices at the level k− j+1 (1 ≤ j ≤ k). Let ∆ ⊆ {1, 2, . . . , k − 1} and F= {Gj : j ∈ ∆}, where Gj is a prescribed weighted graph on each set of children of B at the level k−j+1. In this paper, the eigenvalues of a block symmetric tridiagonal matrix of order n1 +n2 + · · ·+nk are characterized as the ...

Let G be a simple connected graph with pendant vertex set ∂V and nonpendant vertex set V0. The signless Laplacian matrix of G is denoted by Q(G). The signless Dirichlet eigenvalue is a real number λ such that there exists a function f ̸= 0 on V (G) such that Q(G)f(u) = λf(u) for u ∈ V0 and f(u) = 0 for u ∈ ∂V . The signless Dirichlet spectral radius λ(G) is the largest signless Dirichlet eigenva...

for a simple digraph $g$ of order $n$ with vertex set${v_1,v_2,ldots, v_n}$, let $d_i^+$ and $d_i^-$ denote theout-degree and in-degree of a vertex $v_i$ in $g$, respectively. let$d^+(g)=diag(d_1^+,d_2^+,ldots,d_n^+)$ and$d^-(g)=diag(d_1^-,d_2^-,ldots,d_n^-)$. in this paper we introduce$widetilde{sl}(g)=widetilde{d}(g)-s(g)$ to be a new kind of skewlaplacian matrix of $g$, where $widetilde{d}(g...

Let G be a graph with two non adjacent vertices and G′ the graph constructed from G by adding an edge between them. It is known that the trace of Q′ is 2 plus the trace of Q, where Q and Q′ are the signless Laplacian matrices of G and G′ respectively. So, the sum of the Q′-eigenvalues of G′ is the sum of the the Qeigenvalues of G plus two. It is said that Q-spectral integral variation occurs wh...

A spectral graph theory is a theory in which graphs are studied by means of eigenvalues of a matrix M which is in a prescribed way defined for any graph. This theory is called M -theory. We outline a spectral theory of graphs based on the signless Laplacians Q and compare it with other spectral theories, in particular with those based on the adjacency matrix A and the Laplacian L. The Q-theory ...

<abstract><p>Let $ A(G) and D(G) be the adjacency matrix degree diagonal of a graph G $, respectively. For any real number \alpha \in[0, 1] Nikiforov defined A_{\alpha} $-matrix as A_{\alpha}(G) = D(G)+(1-\alpha)A(G) $. Let S_k(A_{\alpha}(G)) sum k largest eigenvalues In this paper, some bounds on are obtained, which not only extends results signless Laplacian matrix, but it also gi...

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

The graph with the largest signless Laplacian spectral radius among all bicyclic graphs with perfect matchings is determined.