Journal of the Ramanujan Mathematical Society

Volume 39, Issue 2, June 2024 pp. 131–142.

On the spectrum of complex unit gain graphs

Authors: Aniruddha Samanta and M. Rajesh Kannan
Author institution:Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur~721~302, India.

Summary: A complex unit gain graph (𝕋-gain graph) Φ = (G, φ) is a graph where the gain function φ assigns a unit complex number to each orientation of an edge of~G, and its inverse is assigned to the opposite orientation. The adjacency matrix A(Φ) of Φ is defined canonically. In~this article, first we study cospectrality of the adjacency matrices of various 𝕋-gain graphs defined on the same underlying graph. Let ρ (Φ) and λ{1}(Φ) be the spectral radius and largest eigenvalue of A(Φ), respectively. A~graph X which contains both directed and undirected edges is known as a mixed graph. Adjacency matrices of mixed graphs are particular cases of adjacency matrices of 𝕋-gain graphs. For any mixed graph~X, the following holds: λ{1}(X) ≤ ρ(X)≤ 3λ{1}(X). We~construct examples to show this inequality need not be true for arbitrary 𝕋-gain graphs. We construct classes of gain graphs for which the above inequality holds. We consider new classes of Hermitian matrices H{k}(X), k = 1,2,…, associated with a mixed graph~X. Finally we establish that ρ(H{k}(X)) ≤ Δ, where Δ is the largest vertex degree of~X, and characterize the structure of X for which the equality holds. As~a consequence, two known results about the spectral radius of adjacency matrices of mixed graphs are deduced.

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