Hermitian Adjacency Matrices of Mixed Graphs

Nullity of Hermitian-Adjacency Matrices of Mixed Graphs

A mixed graph means a graph containing both oriented edges and undirected edges. The nullity of the Hermitian-adjacency matrix of a mixed graph G, denoted by ηH(G), is referred to as the multiplicity of the eigenvalue zero. In this paper, for a mixed unicyclic graph G with given order and matching number, we give a formula on ηH(G), which combines the cases of undirected and oriented unicyclic ...

determinants of adjacency matrices of graphs

we study the set of all determinants of adjacency matrices of graphs with a given number of vertices. using brendan mckay's data base of small graphs, determinants of graphs with at most $9$ vertices are computed so that the number of non-isomorphic graphs with given vertices whose determinants are all equal to a number is exhibited in a table. using an idea of m. newman, it is proved that if $...

Commutativity of the adjacency matrices of graphs

We say that two graphs G1 and G2 with the same vertex set commute if their adjacency matrices commute. In this paper, we find all integers n such that the complete bipartite graph Kn, n is decomposable into commuting perfect matchings or commuting Hamilton cycles. We show that there are at most n−1 linearly independent commuting adjacency matrices of size n; and if this bound occurs, then there...

Determinants of Adjacency Matrices of Graphs

Let G be a simple graph with finite number of vertices. We denote by det(G) the determinant of an adjacency matrix of G. This number det(G) is an integer and is an invariant of G so that its value is independent of the choice of vertices in an adjacency matrix. In this paper, we study the distributions of det(G) whenever G runs over graphs with finite n vertices for a given integer n ≥ 1. We de...

Finding Cliques in Directed Weighted Graphs Using Complex Hermitian Adjacency Matrices