Toward a Laplacian Spectral Determination of Signed ∞-Graphs

Abstract

A signed graph consists of a (simple) graph $G=(V, E)$ together with a function $\sigma:E\rightarrow \{+,-\}$ called signature. Matrices can be associated to signed graphs and the question whether a signed graph is determined by the set of its eigenvalues has gathered the attention of several researchers. In this paper we study the spectral determination with respect to the Laplacian spectrum of signed $\infty$-graphs. After computing some spectral invariants and obtain some constraints on the cospectral mates, we obtain some non isomorphic signed graphs cospectral to signed $\infty$-graphs and we study the spectral characterization of the signed $\infty$-graphs containing a triangle.