arXiv subjects

Zhongyuan Che

Publications and source records attributed to Zhongyuan Che.

9 recordsLinked to original sources

arXiv2026

Isomorphic daisy cubes based on their $τ$-graphs

We prove that if $A$ and $B$ are daisy cubes whose $τ$-graphs are forests, then $A$ and $B$ are isomorphic if and only if their $τ$-graphs are isomorphic. The result is applied to show that a daisy cube with at least one edge is the resonance graph of a plane bipartite graph $G$ if and only if its $τ$-graph is a forest which is isomorphic to the inner dual of the subgraph of $G$ obtained by removing all forbidden edges. As a consequence, some well known properties of Fibonacci cubes and Lucas cubes are provided as examples with different proofs.

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arXiv2026

Resonance graphs that are daisy cubes: from hypercubes to independent sets via resonant sets

Let $G$ be a plane elementary bipartite graph whose infinite face is forcing. We provide a bijection between the set of maximal hypercubes of its resonance graph and the set of maximal resonant sets of $G$, which generalizes a main result in [MATCH Commun. Math. Comput. Chem. 68 (2012) 65-77], where $G$ was only considered as an elementary benzenoid graph without nice coronenes. For a special case when $G$ is a peripherally 2-colorable graph, it follows that there is a bijection between the set of maximal hypercubes of its resonance graph and the set of maximal independent sets of a tree that is the inner dual of $G$. We then show that the resonance graph of a plane bipartite graph $G$ is a daisy cube if and only if it is the simplex graph of the complement of a forest. Finally, we characterize trees with at most 5 maximal independent sets to determine daisy cubes that are simplex graphs of the complements of trees and having at most five maximal vertices.

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arXiv2025

A decomposition structure of resonance graphs that are daisy cubes

It has recently been shown in [\emph{Discrete Appl. Math.} {\bf 366} (2025) 75--85] that the resonance graph of a plane elementary bipartite graph $G$ is a daisy cube if and only if $G$ is peripherally 2-colorable. Let $G$ be a peripherally 2-colorable graph and $R(G)$ be its resonance graph. We provide a decomposition structure of $R(G)$ with respect to an arbitrary finite face of $G$ together with a proper labelling for the vertex set of $R(G)$. An algorithm is obtained to generate a proper labelling for all perfect matchings of $G$ which induces an isometric embedding of $R(G)$ as a daisy cube into an $n$-dimensional hypercube, where $n$ is the isometric dimension of $R(G)$. Moreover, the algorithm can be applied to generate such a proper labelling for all perfect matchings of any plane weakly elementary bipartite graph whose each elementary component with more than two vertices is peripherally 2-colorable. We also compare two binary codings for all perfect matchings of $G$ which induces distinct structures on $R(G)$: one as a daisy cube and the other as a finite distributive, respectively.

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arXiv2024

Resonance graphs of plane bipartite graphs as daisy cubes

We characterize plane bipartite graphs whose resonance graphs are daisy cubes, and therefore generalize related results on resonance graphs of benzenoid graphs, catacondensed even ring systems, as well as 2-connected outerplane bipartite graphs. Firstly, we prove that if $G$ is a plane elementary bipartite graph other than $K_2$, then the resonance graph of $G$ is a daisy cube if and only if the Fries number of $G$ equals the number of finite faces of $G$. Next, we extend the above characterization from plane elementary bipartite graphs to plane bipartite graphs and show that the resonance graph of a plane bipartite graph $G$ is a daisy cube if and only if $G$ is weakly elementary bipartite such that each of its elementary component $G_i$ other than $K_2$ holds the property that the Fries number of $G_i$ equals the number of finite faces of $G_i$. Along the way, we provide a structural characterization for a plane elementary bipartite graph whose resonance graph is a daisy cube, and show that a Cartesian product graph is a daisy cube if and only if all of its nontrivial factors are daisy cubes.

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arXiv2024

Isometric embeddings of resonance graphs as finite distributive lattices

Let $G$ be a plane bipartite graph and $\mathcal{M}(G)$ be the set of all perfect matchings of $G$. The resonance graph $R(G)$ is a graph whose vertex set is $\mathcal{M}(G)$, and two perfect matchings are adjacent in $R(G)$ if their symmetric difference is a cycle forming the periphery of a finite face of $G$. It is known that any connected resonance graph can be isometrically embedded as a finite distributive lattice into hypercubes. The isometric dimension of a connected $R(G)$, denoted by $\mathrm{idim}(R(G))$, is the smallest dimension of a hypercube that $R(G)$ can be isometrically embedded into. Let $d$ be the number of finite faces of $G$ such that there are no forbidden edges on their peripheries. We show that any connected $R(G)$ has $\mathrm{idim}(R(G)) \ge d$ and provide characterizations on when the equality holds. Moreover, if a connected $R(G)$ has $\mathrm{idim}(R(G)) = d$, then we design an algorithm to generate a binary coding of length $d$ for all perfect matchings of $G$ which induces an isometric embedding of $R(G)$ as a finite distributive lattice into a $d$-dimensional hypercube without generating $\mathcal{M}(G)$. Our results provide answers for the fundamental cases of both open questions raised in [\textit{SIAM J. Discrete Math.} {\bf 22} (2008) 971--984.]

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arXiv2023

Outerplane bipartite graphs with isomorphic resonance graphs

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.

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arXiv2019

Peripheral convex expansions of resonance graphs

In this paper, we show that the resonance graph of a plane elementary bipartite graph $G$ can be obtained from an edge by a sequence of peripheral convex expansions with respect to a reducible face decomposition of $G$ if and only if the infinite face of $G$ is forcing.

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arXiv2019

Wiener indices of maximal $k$-degenerate graphs

A graph is maximal $k$-degenerate if each induced subgraph has a vertex of degree at most $k$ and adding any new edge to the graph violates this condition. In this paper, we provide sharp lower and upper bounds on Wiener indices of maximal $k$-degenerate graphs of order $n \ge k \ge 1$. A graph is chordal if every induced cycle in the graph is a triangle and chordal maximal $k$-degenerate graphs of order $n \ge k$ are $k$-trees. For $k$-trees of order $n \ge 2k+2$, we characterize all extremal graphs for the upper bound.

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arXiv2018

An upper bound on the Wiener Index of a k-connected graph

The Wiener index of a connected graph is the summation of all distances between unordered pairs of vertices of the graph. In this paper, we give an upper bound on the Wiener index of a $k$-connected graph $G$ of order $n$ for integers $n-1>k \ge 1$: \[W(G) \le \frac{1}{4} n \lfloor \frac{n+k-2}{k} \rfloor (2n+k-2-k\lfloor \frac{n+k-2}{k} \rfloor).\] Moreover, we show that this upper bound is sharp when $k \ge 2$ is even, and can be obtained by the Wiener index of Harary graph $H_{k,n}$.

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