A note on graphs with purely imaginary per-spectrum
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In 1983, Borowiecki and Jóźwiak 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.
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