arXiv subjects

Shou-Jun Xu

Publications and source records attributed to Shou-Jun Xu.

At least 19 recordsLinked to original sources

arXiv2026

Resolving problems on polynomial characterizations of daisy cubes and extensions

Let $X\subseteq\{0,1\}^n$ be a set of binary strings of length $n$. The daisy cube $Q_n(X)$ is the subgraph of the hypercube $Q_n$ induced by the union of the intervals $I(0^n,x)$ for $x\in X$. As a subclass of partial cubes, it generalizes Fibonacci cubes and Lucas cubes. For a graph $G$ and a vertex $u\in V(G)$, the generating function of the number of $k$-cubes (resp. $k$-cubes at distance $d$ from $u$, and vertices at distance $d$ from $u$) is called the cube polynomial $C_G(x)$ (resp. the distance cube polynomial $D_{G,u}(x,y)$, and the distance polynomial $W_{G,u}(x)$). Let $G$ be a partial cube embedded into the hypercube $Q_n$ with $0^n \in V(G)$. In this paper, we prove that $G$ is a daisy cube if and only if one of the following equivalent conditions holds: (1) $C_{G}(x)=W_{G,0^n}(x+1)$; (2) $D_{G,0^n}(x,y)=W_{G,0^n}(x+y)$; (3) $D_{G,0^n}(x,y)=C_{G}(x+y-1)$. In particular, the results related to (1) and (3) give affirmative answers to two open problems posed by Klavžar and Mollard (2019). Meanwhile, our results yield non-constructive characterizations of daisy cubes, which answer the question posed by Taranenko (2020). Further, we prove that $D_{G, u}(x, y)\leq W_{G, u}(x+y)$ and $C_{G}(x)\leq W_{G, u}(x+1)$ among the whole class of partial cubes. Besides, combined with another sharp upper bound $Cl_{G^\#}(x+1)$ for $C_G(x)$ due to Xie et al.(2024), we obtain polynomial characterizations of simplex graphs (a subclass of daisy cubes): $G$ is a simplex graph if and only if $W_{G, 0^n}(x)=Cl_{G^\#}(x)$, here $Cl_{G^\#}(x)$ is the clique polynomial of the crossing graph $G^\#$ of $G$.

math.CO↗

arXiv2026

Cop numbers for subclasses of partial cubes

The game of Cops and Robbers is a classical pursuit--evasion game on graphs. For a graph $G$, the cop number $c(G)$ is the minimum number of cops needed to guarantee the capture of a robber on $G$. Although this parameter has been determined for several fundamental graph classes, comparatively few exact results are known for partial cubes and their subclasses. We first establish an upper bound for every finite median graph $M$ in terms of its tree-dimension, which improves Crawford and Iršič Chenoweth's bound significantly. This result refines the previous upper bound expressed in terms of a hypercube embedding dimension and can give a substantially smaller estimate. Then we investigate the cop numbers of simplex graphs---a subclass of partial cubes. For a finite graph $G$, the simplex graph $S(G)$ has the cliques of $G$, including the empty clique, as its vertices, with two cliques adjacent whenever they differ in exactly one vertex. We establish a general lower bound for $c(S(G))$ in terms of the clique number of $G$ and a general upper bound in terms of its chromatic number. Finally, as direct applications, we determine the exact values of cop numbers of some special simplex graphs---bipartite wheels, Fibonacci and Lucas cubes.

math.CO↗

arXiv2026

Combinatorial explanation of the weighted Kirchhoff index of graphs

Let $G$ be a connected graph with vertex set $V(G)=\{v_1,v_2,\ldots,v_n\}$, and let $ω:V(G)\to \mathbb R^+$ be a positive vertex-weight function satisfying $ω(v_i)=x_i$ for each $v_i \in V(G)$. The weighted Kirchhoff index of $G$ is defined by $K(G;x_1,x_2,\ldots,x_n)=\sum_{1\le i<j\le n}x_i x_j r_G(v_i,v_j)$, where $r_G(v_i,v_j)$ denotes the resistance distance between $v_i$ and $v_j$. In this paper, we give a combinatorial interpretation of the weighted Kirchhoff index of an arbitrary connected graph. More precisely, we express $K(G;x_1,x_2,\ldots,x_n)$ in terms of the sums of weights of matchings in an appropriately weighted subdivision graph of $G$, and in the subgraphs obtained from this weighted subdivision graph by deleting the subdivision graphs corresponding to $2$-regular subgraphs of $G$. This gives an affirmative answer to a question posed by Li, Li and Yan [Discrete Math. 345 (2022) 113109] concerning a combinatorial explanation of the weighted Kirchhoff index of a general graph by using matchings in weighted subdivision graphs and their subgraphs. As special cases, our formula recovers the known formulas for the weighted Kirchhoff index of trees and unicyclic graphs, as well as the known formula for the ordinary Kirchhoff index of an arbitrary connected graph.

math.CO↗

arXiv2026

Resolving the Klavžar-Kovše conjecture on opposite semicube isomorphisms in partial cubes and its extension

Partial cubes are a fundamental class of graphs that admit isometric embeddings into hypercubes. Klavžar and Kovše [Ars Combin. 93 (2009), 77--86] observed that the opposite semicubes of every harmonic-even partial cube are pairwise isomorphic, and asked whether the converse is true, that is, whether a partial cube is harmonic-even if and only if its opposite semicubes are pairwise isomorphic. In this paper, we answer this question in the negative by constructing an infinite family of partial cubes with pairwise isomorphic opposite semicubes that are not harmonic-even. This establishes that pairwise opposite-semicube isomorphism is strictly weaker than harmonic-evenness and naturally leads to the question of what additional condition restores the equivalence. To address this question, we introduce the opposite-semicube Helly property and prove that a finite partial cube satisfying this property is antipodal, or equivalently harmonic-even by Polat's theorem, if and only if it has pairwise isomorphic opposite semicubes.

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arXiv2025

A solution to Godsil's conjecture on the edge-connectivity of graphs in association schemes

A graph $G$ is called equiarboreal if the number of spanning trees containing a given edge in $G$ is independent of the choice of edge. In [Combinatorica 1(2) (1981) 163--167], Godsil proved that any graph which is a colour class in an association scheme is equiarboreal, and further conjectured that the edge-connectivity of a connected graph which is a colour class in an association scheme equals its vertex degree. In this paper, we confirm this long-standing conjecture. More generally, we prove an even stronger result that the edge-connectivity of a connected regular equiarboreal graph equals its degree by combinatorial and electrical network approaches. As a consequence, we show that every connected regular equiarboreal graph on an even number of vertices has a perfect matching.

math.CO↗

arXiv2025

On the minimum constant resistance curvature conjecture of graphs

Let $G$ be a connected graph with $n$ vertices. The resistance distance $Ω_{G}(i,j)$ between any two vertices $i$ and $j$ of $G$ is defined as the effective resistance between them in the electrical network constructed from $G$ by replacing each edge with a unit resistor. The resistance matrix of $G$, denoted by $R_G$, is an $n \times n$ matrix whose $(i,j)$-entry is equal to $Ω_{G}(i,j)$. The resistance curvature $κ_i$ in the vertex $i$ is defined as the $i$-th component of the vector $(R_G)^{-1}\mathbf{1}$, where $\mathbf{1}$ denotes the all-one vector. If all the curvatures in the vertices of $G$ are equal, then we say that $G$ has constant resistance curvature. Recently, Devriendt, Ottolini and Steinerberger \cite{kde} conjectured that the cycle $C_n$ is extremal in the sense that its curvature is minimum among graphs with constant resistance curvature. In this paper, we confirm the conjecture. As a byproduct, we also solve an open problem proposed by Xu, Liu, Yang and Das \cite{kxu} in 2016. Our proof mainly relies on the characterization of maximum value of the sum of resistance distances from a given vertex to all the other vertices in 2-connected graphs.

math.CO↗

arXiv2025

Characterizing simplex graphs

The simplex graph $S(G)$ of a graph $G$ is defined as the graph whose vertices are the cliques of $G$ (including the empty set), with two vertices being adjacent if, as cliques of $G$, they differ in exactly one vertex. Simplex graphs form a subclass of median graphs and include many well-known families of graphs, such as gear graphs, Fibonacci cubes and Lucas cubes. In this paper, we characterize simplex graphs from four different perspectives: the first focuses on a graph class associated with downwards-closed sets -- namely, the daisy cubes; the second identifies all forbidden partial cube-minors of simplex graphs; the third is from the perspective of the $Θ$ equivalent classes; and the fourth explores the relationship between the maximum degree and the isometric dimension. Furthermore, very recently, Betre et al.\ [K. H. Betre, Y. X. Zhang, C. Edmond, Pure simplicial and clique complexes with a fixed number of facets, 2024, arXiv: 2411.12945v1] proved that an abstract simplicial complex (i.e., an independence system) of a finite set can be represented to a clique complex of a graph if and only if it satisfies the Weak Median Property. As a corollary, we rederive this result by using the graph-theoretical method.

math.CO↗

arXiv2025

Stability of graph pairs involving cycles

A graph pair $(Γ, Σ)$ is called stable if $\aut(Γ)\times\aut(Σ)$ is isomorphic to $\aut(Γ\timesΣ)$ and unstable otherwise, where $Γ\timesΣ$ is the direct product of $Γ$ and $Σ$. A graph is called $R$-thin if distinct vertices have different neighbourhoods. $Γ$ and $Σ$ are said to be coprime if there is no nontrivial graph $Δ$ such that $Γ\cong Γ_1 \times Δ$ and $Σ\cong Σ_1 \times Δ$ for some graphs $Γ_1$ and $Σ_1$. An unstable graph pair $(Γ, Σ)$ is called nontrivially unstable if $Γ$ and $Σ$ are $R$-thin connected coprime graphs and at least one of them is non-bipartite. This paper contributes to the study of the stability of graph pairs with a focus on the case when $Σ= C_n$ is a cycle. We give two sufficient conditions for $(Γ, C_n)$ to be nontrivially unstable, where $n \ne 4$ and $Γ$ is an $R$-thin connected graph. In the case when $Γ$ is an $R$-thin connected non-bipartite graph, we obtain the following results: (i) if $(Γ, K_2)$ is unstable, then $(Γ, C_{n})$ is unstable for every even integer $n \geq 4$; (ii) if an even integer $n \ge 6$ is compatible with $Γ$ in some sense, then $(Γ, C_{n})$ is nontrivially unstable if and only if $(Γ, K_2)$ is unstable; (iii) if there is an even integer $n \ge 6$ compatible with $Γ$ such that $(Γ, C_{n})$ is nontrivially unstable, then $(Γ, C_{m})$ is unstable for all even integers $m \ge 6$. We also prove that if $Γ$ is an $R$-thin connected graph and $n \ge 3$ is an odd integer compatible with $Γ$, then $(Γ, C_{n})$ is stable.

math.CO↗

arXiv2024

A characterization of regular partial cubes whose all convex cycles have the same lengths

Partial cubes are graphs that can be isometrically embedded into hypercubes. Convex cycles play an important role in the study of partial cubes. In this paper, we prove that a regular partial cube is a hypercube (resp., a Doubled Odd graph, an even cycle of length $2n$ where $n\geqslant 4$) if and only if all its convex cycles are 4-cycles (resp., 6-cycles, $2n$-cycles). In particular, the partial cubes whose all convex cycles are 4-cycles are equivalent to almost-median graphs. Therefore, we conclude that regular almost-median graphs are exactly hypercubes, which generalizes the result by Mulder [J. Graph Theory, 4 (1980) 107--110] -- regular median graphs are hypercubes.

math.CO↗

arXiv2024

Ultra log-concavity and real-rootedness of dependence polynomials

For some positive integer $m$, a real polynomial $P(x)=\sum\limits_{k=0}^ma_kx^k$ with $a_k\geqslant 0$ is called log-concave (resp. ultra log-concave) if $a_k^2\geqslant a_{k-1}a_{k+1}$ (resp. $a_k^2\geqslant \left(1+\frac{1}{k}\right)\left(1+\frac{1}{m-k}\right)\cdot$ $a_{k-1}a_{k+1}$) for all $1\leqslant k\leqslant m-1$. If $P(x)$ has only real roots, then it is called real-rooted. It is well-known that the conditions of log-concavity, ultra log-concavity and real-rootedness are ever-stronger. For a graph $G$, a dependent set is a set of vertices which is not independent, i.e., the set of vertices whose induced subgraph contains at least one edge. The dependence polynomial of $G$ is defined as $D(G, x):=\sum\limits_{k\geqslant 0}d_k(G)x^k$, where $d_k(G)$ is the number of dependent sets of size $k$ in $G$. Horrocks proved that $D(G, x)$ is log-concave for every graph $G$ [J. Combin. Theory, Ser. B, 84 (2002) 180--185]. In the present paper, we prove that, for a graph $G$, $D(G, x)$ is ultra log-concave if $G$ is $(K_2\cup 2K_1)$-free or contains an independent set of size $|V(G)|-2$, and give the characterization of graphs whose dependence polynomials are real-rooted. Finally, we focus more attention to the problems of log-concavity about independence systems and pose several conjectures closely related the famous Mason's Conjecture.

math.CO↗

arXiv2024

Maximum values of the edge Mostar index in tricyclic graphs

For a graph $G$, the edge Mostar index of $G$ is the sum of $|m_u(e|G)-m_v(e|G)|$ over all edges $e=uv$ of $G$, where $m_u(e|G)$ denotes the number of edges of $G$ that have a smaller distance in $G$ to $u$ than to $v$, and analogously for $m_v(e|G)$. This paper mainly studies the problem of determining the graphs that maximize the edge Mostar index among tricyclic graphs. To be specific, we determine a sharp upper bound for the edge Mostar index on tricyclic graphs and identify the graphs that attain the bound.

math.CO↗

arXiv2024

Solution to a conjecture on resistance distances of block tower graphs

Let $G$ be a connected graph. The resistance distance between two vertices $u$ and $v$ of $G$, denoted by $R_{G}[u,v]$, is defined as the net effective resistance between them in the electric network constructed from $G$ by replacing each edge with a unit resistor. The resistance diameter of $G$, denoted by $D_{r}(G)$, is defined as the maximum resistance distance among all pairs of vertices of $G$. Let $P_n=a_1a_2\ldots a_n$ be the $n$-vertex path graph and $C_{4}=b_{1}b_2b_3b_4b_{1}$ be the 4-cycle. Then the $n$-th block tower graph $G_n$ is defined as the the Cartesian product of $P_n$ and $C_4$, that is, $G_n=P_{n}\square C_4$. Clearly, the vertex set of $G_n$ is $\{(a_i,b_j)|i=1,\ldots,n;j=1,\ldots,4\}$. In [Discrete Appl. Math. 320 (2022) 387--407], Evans and Francis proposed the following conjecture on resistance distances of $G_n$ and $G_{n+1}$: \begin{equation*} \lim_{n \rightarrow \infty}\left(R_{G_{n+1}}[(a_{1},b_1),(a_{n+1},b_3)]-R_{G_{n}}[(a_{1},b_1),(a_{n},b_3)]\right)=\frac{1}{4}. \end{equation*} In this paper, combining algebraic methods and electrical network approaches, we confirm and further generalize this conjecture. In addition, we determine all the resistance diametrical pairs in $G_n$, which enables us to give an equivalent explanation of the conjecture.

math.CO↗

arXiv2024

A relation between the cube polynomials of partial cubes and the clique polynomials of their crossing graphs

Partial cubes are the graphs which can be embedded into hypercubes. The {\em cube polynomial} of a graph $G$ is a counting polynomial of induced hypercubes of $G$, which is defined as $C(G,x):=\sum_{i\geqslant 0}α_i(G)x^i$, where $α_i(G)$ is the number of induced $i$-cubes (hypercubes of dimension $i$) of $G$. The {\em clique polynomial} of $G$ is defined as $Cl(G,x):=\sum_{i\geqslant 0}a_i(G)x^i$, where $a_i(G)$ ($i\geqslant 1$) is the number of $i$-cliques in $G$ and $a_0(G)=1$. Equivalently, $Cl(G, x)$ is exactly the independence polynomial of the complement $\overline{G}$ of $G$. The {\em crossing graph} $G^{\#}$ of a partial cube $G$ is the graph whose vertices are corresponding to the $Θ$-classes of $G$, and two $Θ$-classes are adjacent in $G^{\#}$ if and only if they cross in $G$. In the present paper, we prove that for a partial cube $G$, $C(G,x)\leqslant Cl(G^{\#}, x+1)$ and the equality holds if and only if $G$ is a median graph. Since every graph can be represented as the crossing graph of a median graph [SIAM J. Discrete Math., 15 (2002) 235--251], the above necessary-and-sufficient result shows that the study on the cube polynomials of median graphs can be transformed to the one on the clique polynomials of general graphs (equivalently, on the independence polynomials of their complements). In addition, we disprove the conjecture that the cube polynomials of median graphs are unimodal.

math.CO↗

arXiv2024

Disproof of a conjecture on the edge Mostar index

For a given connected graph $G$, the edge Mostar index $Mo_e(G)$ is defined as $Mo_e(G)=\sum_{e=uv \in E(G)}|m_u(e|G) - m_v(e|G)|$, where $m_u(e|G)$ and $m_v(e|G)$ are respectively, the number of edges of $G$ lying closer to vertex $u$ than to vertex $v$ and the number of edges of $G$ lying closer to vertex $v$ than to vertex $u$. We determine a sharp upper bound for the edge Mostar index on bicyclic graphs and identify the graphs that attain the bound, which disproves a conjecture proposed by Liu et al. [Iranian J. Math. Chem. 11(2) (2020) 95--106].

math.CO↗

arXiv2024

On regular sets in Cayley graphs

Let $\Ga = (V, E)$ be a graph and $a, b$ nonnegative integers. An $(a, b)$-regular set in $\Ga$ is a nonempty proper subset $D$ of $V$ such that every vertex in $D$ has exactly $a$ neighbours in $D$ and every vertex in $V \setminus D$ has exactly $b$ neighbours in $D$. A $(0,1)$-regular set is called a perfect code, an efficient dominating set, or an independent perfect dominating set. A subset $D$ of a group $G$ is called an $(a,b)$-regular set of $G$ if it is an $(a, b)$-regular set in some Cayley graph of $G$, and an $(a, b)$-regular set in a Cayley graph of $G$ is called a subgroup $(a, b)$-regular set if it is also a subgroup of $G$. In this paper we study $(a, b)$-regular sets in Cayley graphs with a focus on $(0, k)$-regular sets, where $k \ge 1$ is an integer. Among other things we determine when a non-trivial proper normal subgroup of a group is a $(0, k)$-regular set of the group. We also determine all subgroup $(0, k)$-regular sets of dihedral groups and generalized quaternion groups. We obtain necessary and sufficient conditions for a hypercube or the Cartesian product of $n$ copies of the cycle of length $p$ to admit $(0, k)$-regular sets, where $p$ is an odd prime. Our results generalize several known results from perfect codes to $(0, k)$-regular sets.

math.CO↗

arXiv2024

Perfect codes in circulant graphs of degree $p^l-1$

A perfect code in a graph is an independent set of the graph such that every vertex outside the set is adjacent to exactly one vertex in the set. A circulant graph is a Cayley graph of a cyclic group. In this paper we study perfect codes in circulant graphs of degree $p^l - 1$, where $p$ is a prime and $l \ge 1$. We obtain a necessary and sufficient condition for such a circulant graph to admit perfect codes, give a construction of all such circulant graphs which admit perfect codes, and prove a lower bound on the number of distinct perfect codes in such a circulant graph. This extends known results for the case $l=1$ and provides insight on the general problem on the existence and structure of perfect codes in circulant graphs.

math.CO↗

arXiv2023

Solution to an open problem on the closeness of graphs

A network can be analyzed by means of many graph theoretical parameters. In the context of networks analysis, closeness is a structural metric that evaluates a node's significance inside a network. A cactus is a connected graph in which any block is either a cut edge or a cycle. This paper analyzes the closeness of cacti, we determine the unique graph that minimizes the closeness over all cacti with fixed numbers of vertices and cycles, which solves an open problem proposed by Poklukar \& Žerovnik [Fundam. Inform. 167 (2019) 219--234].

cs.SI↗

arXiv2023

Extremal results on the Mostar index of trees with fixed parameters

For a graph $G$, the Mostar index of $G$ is the sum of $|n_u(e)$ - $n_v(e)|$ over all edges $e=uv$ of $G$, where $n_u(e)$ denotes the number of vertices of $G$ that have a smaller distance in $G$ to $u$ than to $v$, and analogously for $n_v(e)$. We determine all the graphs that maximize and minimize the Mostar index respectively over all trees in terms of some fixed parameters like the number of odd vertices, the number of vertices of degree two, and the number of pendent paths of fixed length.

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