Let G be a graph and let A and D be the adjacency matrix of G and diagonal matrix of vertex degrees of G respectively. If each vertex degree is positive, then the normalized adjacency matrix of G is  = D−1/2AD−1/2. A classification is given of those graphs for which the all eigenvalues of the normalized adjacency matrix are integral. The problem of determining those graphs G for which λ ∈ Q fo...

We present the characteristic polynomial for transition matrix of a vertex-face walk on graph, and obtain its spectra. Furthermore, we express 2-dimensional torus by using adjacency matrix, As an application, define new walk-type zeta function with respect to two-dimensional torus, explicit formula.

A mixed graph is said to be second kind hermitian integral (HS-integral) if the eigenvalues of its Hermitian-adjacency matrix are integers. called Eisenstein (0, 1)-adjacency We characterize set S for which normal Cayley Cay(Γ,S) HS-integral any finite group Γ. further show that a and only it integral. This paper generalizes results Kadyan Bhattacharjy (2022) [11].

This is the third part of our work with a common title. The first [11] and the second part [12] will be also referred in the sequel as Part I and Part II, respectively. This third part was not planned at the beginning, but a lot of recently published papers on the signless Laplacian eigenvalues of graphs and some observations of ours justify its preparation. By a spectral graph theory we unders...

A matrix-weighted graph is an undirected with a k×k symmetric positive semidefinite matrix assigned to each edge. Such graphs admit natural generalizations of the Laplacian and adjacency matrices, leading generalized notion expansion. Extensions some theorems about expansion hold for graphs—in particular, analogue expander mixing lemma one half Cheeger-type inequality. These results lead defini...

The use of spectral methods in graph theory has allowed for some amazing results where an arithmetic invariant (i.e., diameter, chromatic number, and so on) has been bounded and analyzed using analytic tools. The key has been to examine the spectrum of various matrices associated with graphs and to try to “hear the shape” of the graph from the spectrum. The three most widely used spectrums are ...

let $n$ be any positive integer, the friendship graph $f_n$ consists of $n$ edge-disjoint triangles that all of them meeting in one vertex. a graph $g$ is called cospectral with a graph $h$ if their adjacency matrices have the same eigenvalues. recently in href{http://arxiv.org/pdf/1310.6529v1.pdf}{http://arxiv.org/pdf/1310.6529v1.pdf} it is proved that if $g$ is any graph cospectral with $f_n$...

The spectrum of a graph G is the set of eigenvalues of the 0–1 adjacency matrix of G. The nullity of a graph is the number of zeros in its spectrum. It is shown that the nullity of the line graph of a tree is at most one. c © 2001 Elsevier Science B.V. All rights reserved.

Abstract: The structure of bus network is very significant for bus system. To evaluate the performance of the structure of bus network, indicators basing on graph theory and complex network theory are proposed. Three forms of matrices comprising linestation matrix, weighted adjacency matrix and adjacency matrix under space P are used to represent the bus network. The paper proposes a shift powe...

Problem 1. [20 points] The adjacency matrix A of a graph G with n vertices as defined in lecture is an n×n matrix in which Ai,j is 1 if there is an edge from i to j and 0 if there is not. In lecture we saw how the smallest k where Ai,j 6= 0 describes the length of the shortest path from i to j. Given a combinatorial interpretation of the following statements about the adjacency matrix in terms ...