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.