NEPS of complex unit gain graphs

Abstract

A complex unit gain graph (or $\mathbb T$-gain graph) is a gain graph with gains in $\mathbb T$, the multiplicative group of complex units. Extending a classical construction for simple graphs due to Cvektovic, suitably defined noncomplete extended $p$-sums (NEPS, for short) of $\mathbb T$-gain graphs are considered in this paper. Structural properties of NEPS like balance and some spectral properties and invariants of their adjacency and Laplacian matrices are investigated, including the energy and the possible symmetry of the adjacency spectrum. It is also shown how NEPS are useful to obtain infinitely many integral graphs from the few at hands.
Moreover, it is studied how NEPS of $\mathbb T$-gain graphs behave with respect to the property of being nut, i.e., having $0$ as simple adjacency eigenvalue and nowhere zero $0$-eigenvectors. Finally, a family of new products generalizing NEPS is introduced, and their few first spectral properties explored.