Integral regular net-balanced signed graphs with vertex degree at most four

Abstract

A signed graph is called integral if its spectrum consists entirely of integers, it is r -regular if its underlying graph is regular of degree r , and it is net-balanced if the difference between positive and negative vertex degree is a constant on the vertex set (this constant is called the net-balance and denoted ρ ). We determine all the connected integral 3 -regular net-balanced signed graphs. In the next natural step, for r = 4 , we consider only those whose net-balance is a simple eigenvalue. There, we complete the list of feasible spectra in bipartite case for ρ ≠ 0 and prove the non-existence for ρ = 0 . Certain existence conditions are established and the existence of some 4 -regular (simple) graphs is confirmed. In this study we transferred some results from the theory of graph spectra; in particular, we give a counterpart to the Hoffman polynomial.