A note on graphs with purely imaginary per-spectrum

Abstract

In 1983, Borowiecki and Jóźwiak 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 K2,3, 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 K2,3 (planar and nonplanar) using coalescence of rooted graphs.