Signed graphs cospectral with the path

TL;DR

This paper investigates when the spectrum of a signed path graph uniquely determines its structure, establishing specific conditions based on the path length modulo 4 and exceptions.

Contribution

It proves that signed paths are determined by their spectrum for most lengths, with specific exceptions, advancing understanding of spectral graph characterization.

Findings

Signed paths are spectrally determined for most lengths.

Specific exceptions for path lengths include 3, 8, 13, 14, 17, 29, and those congruent to 3 mod 4.

Spectral characterization depends on path length modulo 4 and certain exceptions.

Abstract

A signed graph $\Gamma$ is said to be determined by its spectrum if every signed graph with the same spectrum as $\Gamma$ is switching isomorphic with $\Gamma$. Here it is proved that the path $P_n$, interpreted as a signed graph, is determined by its spectrum if and only if $n\equiv 0, 1$, or 2 (mod 4), unless $n\in\{8, 13, 14, 17, 29\}$, or $n=3$.