Ela Spectra of Weighted Rooted Graphs Having Prescribed Subgraphs at Some Levels
نویسندگان
چکیده
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 eigenvalues of symmetric tridiagonal matrices of order j, 1 ≤ j ≤ k, easily constructed from the degrees of the vertices, the weights of the edges, and the eigenvalues of the matrices associated to the family of graphs F. These results are applied to characterize the eigenvalues of the Laplacian matrix, including their multiplicities, of the graph B (F) obtained from B and all the graphs in F = {Gj : j ∈ ∆} ; and also of the signless Laplacian and adjacency matrices whenever the graphs of the family F are regular.
منابع مشابه
Ela Copies of a Rooted Weighted Graph Attached to an Arbitrary Weighted Graph and Applications
The spectrum of the Laplacian, signless Laplacian and adjacency matrices of the family of the weighted graphs R{H}, obtained from a connected weighted graph R on r vertices and r copies of a modified Bethe tree H by identifying the root of the i-th copy of H with the i-th vertex of R, is determined.
متن کاملOn Graphs with Adjacent Vertices of Large Degree
Let g(n, m) denote the class of simple graphs on n vertices and m edges and let C E 9( a, m). For suitably restricted values of m, C will necessarily contain certain prescribed subgraphs such as cycles of given lengths and complete graphs . For example> if m > 7nZ then G contains cycles of all lengths up to Li(n+ 3) J . Recently we have established a number of results concerning the existence o...
متن کاملEla on Spectra of Expansion Graphs and Matrix Polynomials
An expansion graph of a directed weighted graph G0 is obtained fromG0 by replacing some edges by disjoint chains. The adjacency matrix of an expansion graph is a partial linearization of a matrix polynomial with nonnegative coefficients. The spectral radii for different expansion graphs of G0 and correspondingly the spectral radii of matrix polynomials with nonnegative coefficients, which sum u...
متن کاملTOTAL DOMINATION POLYNOMIAL OF GRAPHS FROM PRIMARY SUBGRAPHS
Let $G = (V, E)$ be a simple graph of order $n$. The total dominating set is a subset $D$ of $V$ that every vertex of $V$ is adjacent to some vertices of $D$. The total domination number of $G$ is equal to minimum cardinality of total dominating set in $G$ and denoted by $gamma_t(G)$. The total domination polynomial of $G$ is the polynomial $D_t(G,x)=sum d_t(G,i)$, where $d_t(G,i)$ is the numbe...
متن کامل