Spectral analysis of t-path signed graphs

Formally, a signed graph S is a pair (G,σ) that consists of a graph G=(V,E) and a sign mapping called signature σ from E to the sign group {+, -}. Given a signed graph S and a positive integer t, the t-path signed graph (S)t of S is a signed graph whose vertex set is V(S) and two vertices are adjacent if and only if there exists a path of length t between these vertices and then by defining its sign st(e) to be ‘-’ if and only if in every such path of length t in S all the edges are negative. The negation η(S) of a signed graph S is a signed graph obtained from S by reversing the sign of every edge of S. Two signed graphs S1 and S2 on the same underlying graph are switching equivalent if it is possible to assign signs ‘+’ (‘plus’) or ‘-’ (‘minus’) to the vertices of S1 such that by reversing the sign of each of its edges that have received opposite signs at its ends, one obtains S2. In this paper, we characterize signed graphs whose negations are switching equivalent to their t-path signed graphs for t=2 and also characterize signed graphs such that the spectrum of their t-path signed graphs, where t=1, and 2, is symmetric about the origin.