A note on graphs with purely imaginary per-spectrum

DOI10.1016/J.AMC.2024.128754arXiv2211.13072MaRDI QIDQ6507139

Hitesh Wankhede, Ranveer Singh

Abstract: In 1983, Borowiecki and J'o'zwiak posed an open problem of characterizing graphs with purely imaginary per-spectrum. The most general result, although a partial solution, was given in 2004 by Yan and Zhang, who show that if a graph contains 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.

Mathematics Subject Classification ID

Trees (05C05) Graph polynomials (05C31) Graphs and linear algebra (matrices, eigenvalues, etc.) (05C50) Graph operations (line graphs, products, etc.) (05C76)

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