Integral regular net-balanced signed graphs with vertex degree at most four - Stanić, Zoran (Ars mathematica contemporanea, 2019)
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.
Keywords: adjacency matrix; matrika sosednosti; net-balanced signed graph; neto-uravnoteženi predznačeni grafi; predznačeni graf; preklopno ekvivalentni predznačeni grafi; signed graph; switching equivalent signed graphs