Core Concepts
Graphs with purely imaginary per-spectrum can be constructed by coalescing a rooted graph and a rooted tree, where the roots satisfy certain conditions.
Abstract
The article discusses the problem of characterizing graphs with purely imaginary per-spectrum, which was posed by Borowiecki and Jóźwiak in 1983. The authors provide a construction method for such graphs using the coalescence of rooted graphs.
Key highlights:
Bipartite graphs containing no subgraph which is an even subdivision of K2,3 have purely imaginary per-spectrum. Such graphs are planar and admit a Pfaffian orientation.
The authors investigate the case where the bipartite graph contains a subgraph which is an even subdivision of K2,3.
They provide a construction method where the coalescence of a rooted graph (G1, r1) and a rooted tree (T, r2) results in a graph with purely imaginary per-spectrum, if the polynomial H(G1, T) has all its roots purely imaginary.
The authors demonstrate the construction for specific cases, such as when G1 is K2,3 rooted at a degree 3 vertex or a degree 2 vertex, and T is a starlike or pathlike tree.
They also provide examples of nonplanar graphs with purely imaginary per-spectrum obtained by coalescing K3,3 with rooted starlike or pathlike trees.
The article concludes by discussing various ways to extend the construction idea, such as using different rooted graphs or different definitions of the rooted product.