Spectral analysis of t-path signed graphs

Abstract

Formally, a signed graph S is a pair ($G,\sigma$) that consists of a graph $G=(V,E)$ and a sign mapping called signature $\sigma$ 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 $s_t(e)$ to be ‘-’ if and only if in every such path of length t in S all the edges are negative. The negation $\eta (S)$ of a signed graph S is a signed graph obtained from S by reversing the sign of every edge of S. Two signed graphs $S_1$ and $S_2$ on the same underlying graph are switching equivalent if it is possible to assign signs ‘$+$’ (‘plus’) or ‘$-$’ (‘minus’) to the vertices of $S_1$ such that by reversing the sign of each of its edges that have received opposite signs at its ends, one obtains $S_2$. 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.

Keywords:

• Balanced signed graph
• marked signed graph
• signed isomorphism
• switching equivalence
• t-path signed graph
• spectrum of a matrix
• eigenvalues

AMS Subject Classifications:

• 05C22
• 05C75

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by University Grant Commission [Sr. No. 2061540883 Ref. No. 21/06/2015(i)EU-V].