A note on graphs with purely imaginary per-spectrum

TL;DR

This paper explores the construction of graphs with purely imaginary per-spectrum, extending known results by including graphs with certain subdivisions of K_{2,3} through coalescence methods.

Contribution

It introduces a new construction technique for graphs with purely imaginary per-spectrum, including those with an even subdivision of K_{2,3}, regardless of planarity.

Findings

Constructed graphs with purely imaginary per-spectrum using coalescence.

Extended the class of known graphs with purely imaginary per-spectrum.

Included both planar and nonplanar graphs with specific subgraph structures.

Abstract

In 1983, Borowiecki and J\'o\'zwiak posed the problem ``Characterize those graphs which have purely imaginary per-spectrum.'' This problem is still open. The most general result, although a partial solution, was given in 2004 by Yan and Zhang, who show that if $G$ is a bipartite graph containing no subgraph which is an even subdivision of $K_{2,3}$, then it has purely imaginary per-spectrum. Zhang and Li in 2012 proved that such graphs are planar and admit a Pfaffian orientation. In this article, we describe how to construct graphs with purely imaginary per-spectrum having a subgraph which is an even subdivision of $K_{2,3}$ (planar and nonplanar) using coalescence of rooted graphs.