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A note on graphs with purely imaginary per-spectrum

Ranveer Singh, Hitesh Wankhede

TL;DR

The paper addresses the problem of characterizing graphs with purely imaginary per-spectrum by developing a coalescence-based construction that combines a rooted graph with a rooted tree and yields a test polynomial to ensure purely imaginary roots. The authors derive a general permanental-polynomial recurrence for coalesced graphs and define the auxiliary polynomial $\mathcal{H}(G_1,T)$ to determine when the resulting graph lies in the target class, applying the method to coalescences involving $K_{2,3}$ and $K_{3,3}$ with rooted starlike or pathlike trees. They provide explicit parameter regimes (e.g., for $G_1=K_{2,3}$) and identify minimal planar ($G_8$) and nonplanar ($G_{11}$) examples, illustrating the practical reach of the construction. The work expands constructive techniques for graphs with purely imaginary permanental spectra, connecting to known results on Pfaffian orientations and planarity while outlining avenues for generalization and further study.

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 $K_{2,3}$, 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.

A note on graphs with purely imaginary per-spectrum

TL;DR

The paper addresses the problem of characterizing graphs with purely imaginary per-spectrum by developing a coalescence-based construction that combines a rooted graph with a rooted tree and yields a test polynomial to ensure purely imaginary roots. The authors derive a general permanental-polynomial recurrence for coalesced graphs and define the auxiliary polynomial to determine when the resulting graph lies in the target class, applying the method to coalescences involving and with rooted starlike or pathlike trees. They provide explicit parameter regimes (e.g., for ) and identify minimal planar () and nonplanar () examples, illustrating the practical reach of the construction. The work expands constructive techniques for graphs with purely imaginary permanental spectra, connecting to known results on Pfaffian orientations and planarity while outlining avenues for generalization and further study.

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 is a bipartite graph containing no subgraph which is an even subdivision of , 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 (planar and nonplanar) using coalescence of rooted graphs.
Paper Structure (4 sections, 13 theorems, 21 equations, 5 figures)

This paper contains 4 sections, 13 theorems, 21 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Construction
  4. Conclusion

Key Result

Proposition 2.1

borowiecki1980computing Let $G$ be a graph on $n$ vertices with $\pi(G,x)=\sum_{k=0}^{n}b_kx^{n-k}$. Then, where the summation is taken over all Sachs subgraphs $U_k$ on $k$ vertices of $G$, and $c(U_k)$ denotes the number of components of $U_k$ which are cycles.

Figures (5)

  • Figure 1:
  • Figure 2: $\mathcal{B}$: Bipartite Graphs, $\mathcal{P}$: Planar Bipartite Graphs, $\mathcal{G}$: Graphs with purely imaginary per-spectrum, $\mathcal{K}$: Graphs without a subgraph which is an even subdivision of $K_{2,3}$.
  • Figure 3: (Left to right) $K_{2,4}$, $G_{8}$, $K_{3,3}$ and $G_{11}$. By subdividing the blue edge twice in $K_{2, 4}$ we obtain $G_8$. By subdividing the blue and the red edge twice in $K_{3, 3}$, and drawing an edge between their common vertex and the newly added vertex, we obtain $G_{11}$.
  • Figure 4: Connected bipartite graphs on $9$ vertices in $\mathcal{G}$ which have an even subdivision of $K_{2,3}$ as a subgraph
  • Figure 5: Scatter plot of points $(l,k)$ where the constructed graphs have purely imaginary per-spectrum.

Theorems & Definitions (26)

  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • Theorem 2.4
  • Lemma 2.5
  • Proposition 2.6
  • proof
  • Proposition 2.7
  • proof
  • ...and 16 more