# 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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