Signed graphs with integral net Laplacian spectrum

Abstract

AbstractGiven a signed graph [Formula: see text], let [Formula: see text] and [Formula: see text] be its standard adjacency matrix and the diagonal matrix of net-degrees, respectively. The net Laplacian matrix of [Formula: see text] is defined as [Formula: see text]. 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: see text] are simple and the other in which [Formula: see text] 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.

Keywords

• Net-degree
• net Laplacian matrix
• join
• signed cograph
• graph product
• regular signed graph