Signed graphs with integral net Laplacian spectrum

Abstract

Given a signed graph (Formula presented.), let (Formula presented.) and (Formula presented.) be its standard adjacency matrix and the diagonal matrix of net-degrees, respectively. The net Laplacian matrix of (Formula presented.) is defined as (Formula presented.). In this paper we investigate signed graphs whose net Laplacian spectrum consists entirely of integers. The focus is mainly on the two extreme cases, the one in which all eigenvalues of (Formula presented.) are simple and the other in which (Formula presented.) has 2 or 3 (distinct) eigenvalues. Both cases include structure theorems, degree constraints and particular constructions of new examples. Several applications in the framework of control theory are reported.