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Outerplane bipartite graphs with isomorphic resonance graphs

Simon Brezovnik, Zhongyuan Che, Niko Tratnik, Petra Žigert Pleteršek

TL;DR

This work resolves the problem of characterizing when resonance graphs of two $2$-connected outerplane bipartite graphs are isomorphic. By leveraging reducible-face decompositions, peripheral convex expansions, and the interplay between resonance graphs, resonance digraphs, and the inner dual, the authors derive a structure-based criterion that is both necessary and sufficient. They show that resonance-graph isomorphism is equivalent to either isomorphism of resonance digraphs under a proper two-coloring or to an inner-dual isomorphism that preserves the regularity of all $3$-paths in the inner dual. The results yield concrete corollaries and provide a framework with potential extensions to broader graph families, advancing the understanding of resonance graphs in chemical and mathematical contexts.

Abstract

We present novel results related to isomorphic resonance graphs of 2-connected outerplane bipartite graphs. As the main result, we provide a structure characterization for 2-connected outerplane bipartite graphs with isomorphic resonance graphs. Moreover, two additional characterizations are expressed in terms of resonance digraphs and via local structures of inner duals of 2-connected outerplane bipartite graphs, respectively.

Outerplane bipartite graphs with isomorphic resonance graphs

TL;DR

This work resolves the problem of characterizing when resonance graphs of two -connected outerplane bipartite graphs are isomorphic. By leveraging reducible-face decompositions, peripheral convex expansions, and the interplay between resonance graphs, resonance digraphs, and the inner dual, the authors derive a structure-based criterion that is both necessary and sufficient. They show that resonance-graph isomorphism is equivalent to either isomorphism of resonance digraphs under a proper two-coloring or to an inner-dual isomorphism that preserves the regularity of all -paths in the inner dual. The results yield concrete corollaries and provide a framework with potential extensions to broader graph families, advancing the understanding of resonance graphs in chemical and mathematical contexts.

Abstract

We present novel results related to isomorphic resonance graphs of 2-connected outerplane bipartite graphs. As the main result, we provide a structure characterization for 2-connected outerplane bipartite graphs with isomorphic resonance graphs. Moreover, two additional characterizations are expressed in terms of resonance digraphs and via local structures of inner duals of 2-connected outerplane bipartite graphs, respectively.
Paper Structure (3 sections, 11 theorems, 1 equation, 2 figures)

This paper contains 3 sections, 11 theorems, 1 equation, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results

Key Result

Proposition 2.1

TV12 Let $G$ be a plane elementary bipartite graph other than $K_2$. Then the outer cycle of $G$ is improper $M_{\hat{0}}$-alternating as well as proper $M_{\hat{1}}$-alternating, where $M_{\hat{0}}$ is the minimum and $M_{\hat{1}}$ the maximum in the finite distributive lattice $\mathcal{M}(G)$.

Figures (2)

  • Figure 1: Resonance graphs of the linear benzenoid chain and fibonaccene with three hexagons.
  • Figure 2: A peripheral convex expansion of the resonance graph $R(G)$.

Theorems & Definitions (15)

  • Proposition 2.1
  • Theorem 2.2
  • Proposition 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • ...and 5 more