On the spectrum of complex unit gain graph

Abstract

A T-gain graph is a simple graph in which a unit complex number is assigned to each orientation of an edge, and its inverse is assigned to the opposite orientation. The associated adjacency matrix is defined canonically, and is called T-gain adjacency matrix. Let T G denote the collection of all T-gain adjacency matrices on a graph G. In this article, we study the cospectrality of matrices in T G and we establish equivalent conditions for a graph G to be a tree in terms of the spectrum and the spectral radius of matrices in T G . We identify a class of connected graphs F such that for each G ∈ F , the matrices in T G have nonnegative real part up to diagonal unitary similarity. Then we establish bounds for the spectral radius of T-gain adjacency matrices on G ∈ F in terms of their largest eigenvalues. Thereupon, we characterize T-gain graphs for which the spectral radius of the associated T-gain adjacency matrices equal to the largest vertex degree of the underlying graph. These bounds generalize results known for the spectral radius of Hermitian adjacency matrices of digraphs and provide an alternate proof of a result about the sharpness of the bound in terms of largest vertex degree established in [Krystal Guo, Bojan Mohar. Hermitian adjacency matrix of digraphs and mixed graphs.

Switching equivalence of Hermitian adjacency matrix of mixed graphs

“…In In the next two results, we establish two sufficient conditions under which ℜ A (C) = ℜ B (C) for all cycles C in G A whenever A and B are cospectral. Some similar results appear in [10]. Our proof technique is also similar with that in [10].…”

“…The seminal paper by Collatz and Sinogowitz [3] provided the first examples of cospectral trees. Cospectral graphs are studied for signed graphs [2], oriented graphs [4], mixed graphs [8], complex unit gain graph [10] etc. In all these graphs, to characterize the cospectral graphs, researchers introduced a switching equivalence relation, which is a similarity transformation.…”

Spectral Fundamentals and Characterizations of Signed Directed Graphs

“…A well-known tool in spectral graph theory is known as the Harary-Sachs coefficients theorem [4], which computes the coefficients of the characteristic polynomial of a graph based on some structural properties of its elementary spanning subgraphs. For the current work, we may specialize a result by Samanta and Kannan [17], to be significantly more combinatorially approachable. We first impose some simple notation.…”

“…Similar operations are used for the Hermitian adjacency matrix [8] and for gain graph context [15,17]. The final operation under which equivalence classes should realistically be closed stems from algebraic number theory; specifically, the Galois automorphism [5] over the quadratic ring Z[ω].…”

“…At the same time, various authors (see, e.g., [11,15,17]) have been considering complex unit gain graphs. Let T = {z ∈ C : |z|= 1} be the multiplicative group of unit complex numbers.…”