arXiv subjects

Kexiang Xu

Publications and source records attributed to Kexiang Xu.

At least 19 recordsLinked to original sources

arXiv2026

Counting oriented spanning trees in generalized join digraphs

Let $G$ be a digraph with vertex set $\{1,2,...,n\}$ and $H_{1},H_{2},...,H_{n}$ be $n$ digraphs. The generalized join digraph $\overrightarrow{G}=G[H_{1},H_{2},...,H_{n}]$ is a digraph obtained from $G$ by replacing each vertex $i$ with $H_{i}$ and for any $u\in V(H_{i})$ and $v\in V(H_{j})$, $(u,v)\in E(\overrightarrow{G})$ if and only if $(i,j)\in E(G)$. In this paper we express the number of oriented spanning trees in $\overrightarrow{G}$ in terms of Laplacian eigenvalues of $H_{1},H_{2},...,H_{n}$ and oriented spanning trees of $G$. Furthermore, we consider the number of oriented spanning trees with a fixed root in $\overrightarrow{G}$. First, we introduce the biclique-directed star transformation formula for counting oriented spanning trees with a fixed root in digraphs. Using it, we give the formula for the total number of oriented spanning trees with roots in a certain $H_{i}$ $(1\leq i \leq n)$ of $\overrightarrow{G}$ in terms of Laplacian eigenvalues of $H_{1},H_{2},...,H_{n}$ and oriented spanning trees of $G$. As applications, when each $H_{i}$ is a given digraph, the enumerative formulas for oriented spanning trees with a fixed root of $\overrightarrow{G}$ are derived from our work.

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arXiv2026

Edge transmission irregular graphs

The transmission of a vertex $v$ in a connected graph $G$ is the sum of distances from $v$ to all vertices in $G$. A transmission irregular (TI) graph is a connected graph in which any two distinct vertices have different transmissions. We extend the concept of transmission to edges by defining the transmission of an edge as the sum of the transmissions of its two endpoints. A connected graph can now be called edge transmission irregular (ETI) if any two distinct edges have different transmissions. We show that almost all graphs are not ETI and then investigate several related order realizability problems involving chemical ETI graphs. In particular, we prove that for every $n \ge 15$, there exists a subcubic tree of order $n$ that is both TI and ETI.

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arXiv2026

On asymptotic values for the minimum number of spanning forests in simple regular graphs

Let $F(G)$ be the number of spanning forests in a graph $G$ and $\mathcal{C}(n,d)$ be the set of all connected $d$-regular simple graphs of order $n$. Define $\widehat{f}_{d}=\liminf_{n\rightarrow \infty}\{F(G)^{1/n}:G\in \mathcal{C}(n,d)\}$. Let $n_i$ be the number of vertices of degree $i$ in $G$. In this paper we give two lower bounds for $F(G)$ in terms of $n_i$ in connected graphs whose vertex degrees belong to $\{2,3\}$ and $\{2,3,4\}$, respectively. Furthermore, we determine the exact values of $\widehat{f}_3$ and $\widehat{f}_4$.

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arXiv2026

On the Variance Fraction of the Hard-Core Model on Graphs with Bounded Maximum Degree

The hard-core model can be used to understand the number of independent sets in graphs in extremal graph theory. The occupancy fraction, defined by Davies \textit{et al.} in 2017 as the logarithmic derivative of the independence polynomial of a graph, is a key quantity in the hard-core model. The variance fraction, introduced by Davies \textit{et al.} in 2025, is defined as the derivative of the occupancy fraction with respect to the logarithm of the fugacity. Since the occupancy fraction can be obtained by integrating the variance fraction with respect to the logarithm of the fugacity, bounding the variance fraction yields the corresponding bounds on the occupancy fraction. Moreover, the occupancy fraction correlates, in quantity, to the independence polynomial. In this note we provide two lower bounds on the variance fraction, proving the conjecture by Davies \textit{et al.} in 2025, for graphs with bounded maximum degree and for graphs with $n$ vertices, respectively. We also derive lower bounds for other graph classes, including graphs with a given edge chromatic number, $d$-regular graphs, and triangle-free graphs with bounded maximum degree.

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arXiv2026

The enumeration of odd spanning trees in graphs

A graph is odd if all of its vertices have odd degrees. In particular, an odd spanning tree in a connected graph is a spanning tree in which all vertices have odd degrees. In this paper we establish a unified technique to enumerate odd spanning trees of a graph $G$ in terms of a multivariable polynomial associated with $G$ and indeterminates $\{x_{i}:v_i\in V(G)\}$. As applications, the enumerative formulas for odd spanning trees in complete graphs, complete multipartite graphs, almost complete graphs, complete split graphs and Ferrers graphs are, respectively, derived from our work.

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arXiv2025

On the transmission irregular trees with the maximum Wiener index

The transmission of a vertex $v$ in a (chemical) graph $G$ is the sum of distances from $v$ to other vertices in $G$. If any two vertices of $G$ have different transmissions, then $G$ is transmission irregular. The Wiener index $W(G)$ of a graph $G$ is the sum of all distances between all unordered pairs of vertices in $G$, which has another formula as the half of the sum of transmissions of all vertices of $G$. In this paper, we consider the Wiener index maximization problem on the set of transmission irregular trees of a given order $n \in \mathbb{N}$. We solve the problem for all odd values of $n$ and for almost all even values of $n$. Each resolved extremal problem has a unique solution that is a chemical tree.

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arXiv2025

Maximum number of spanning trees and connectivity: Graphs with a fixed minimum degree and bipartite graphs

The number of spanning trees in a graph $G$ is the total number of distinct spanning subgraphs of $G$ that are trees. In this paper we characterize the unique graph with a prescribed vertex (resp. edge) connectivity, minimum degree and order that attains the maximum number of spanning trees. Moreover, all the bipartite graphs are determined with a given vertex (resp. edge) connectivity and order maximizing the number of spanning trees.

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arXiv2024

Uniquely $C_{4}^{+}$-saturated graphs

A graph $G$ is uniquely $H$-saturated if it contains no copy of a graph $H$ as a subgraph, but adding any new edge into $G$ creates exactly one copy of $H$. Let $C_{4}^{+}$ be the diamond graph consisting of a $4$-cycle $C_{4}$ with one chord and $C_{3}^{*}$ be the graph consisting of a triangle with a pendant edge. In this paper we prove that a nontrivial uniquely $C_{4}^{+}$-saturated graph $G$ has girth $3$ or $4$. Further, $G$ has girth $4$ if and only if it is a strongly regular graph with special parameters. For $n>18k^{2}-24k+10$ with $k\geq2$, there are no uniquely $C_{4}^{+}$-saturated graphs on $n$ vertices with $k$ triangles. In particular, $C_{3}^{*}$ is the only nontrivial uniquely $C_{4}^{+}$-saturated graph with one triangle, and there are no uniquely $C_{4}^{+}$-saturated graphs with two, three or four triangles.

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arXiv2023

Solving the Mostar index inverse problem

A nonnegative integer $p$ is realizable by a graph-theoretical invariant $I$ if there exist a graph $G$ such that $I(G) = p$. The inverse problem for $I$ consists of finding all nonnegative integers $p$ realizable by $I$. In this paper, we consider and solve the inverse problem for the Mostar index, a recently introduced graph-theoretical invariant which attracted a lot of attention in recent years in both the mathematical and the chemical community. We show that a nonnegative integer is realizable by the Mostar index if and only if it is not equal to one. Besides presenting the complete solution to the problem, we also present some empirical observations and outline several open problems and possible directions for further research.

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arXiv2022

New transmission irregular chemical graphs

The transmission of a vertex $v$ of a (chemical) graph $G$ is the sum of distances from $v$ to other vertices in $G$. If any two vertices of $G$ have different transmissions, then $G$ is a transmission irregular graph. It is shown that for any odd number $n\geq 7$ there exists a transmission irregular chemical tree of order $n$. A construction is provided which generates new transmission irregular (chemical) trees. Two additional families of chemical graphs are characterized by property of transmission irregularity and two sufficient condition provided which guarantee that the transmission irregularity is preserved upon adding a new edge.

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arXiv2022

On the general position numbers of maximal outerplanar graphs

A subset $R\subseteq V(G)$ of a graph $G$ is a general position set if any triple set $R_0$ of $R$ is non-geodesic in $G$, that is, no vertex of $R_0$ lies on any geodesic between the other two vertices of $R_0$ in $G$. Let $\mathcal{R}$ be the set of general position sets of a graph $G$. The general position number of a graph $G$, denoted by $gp(G)$, is defined as $gp(G)=\max\{|R|:R\in\mathcal{R}\}$. In this paper, we determine the bounds on the gp-numbers for any maximal outerplane graph and characterize the corresponding extremal graphs.

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arXiv2022

Relating the total domination number and the annihilation number for quasi-trees and some composite graphs

The total domination number $γ_{t}(G)$ of a graph $G$ is the cardinality of a smallest set $D\subseteq V(G)$ such that each vertex of $G$ has a neighbor in $D$. The annihilation number $a(G)$ of $G$ is the largest integer $k$ such that there exist $k$ different vertices in $G$ with the degree sum at most $m(G)$. It is conjectured that $γ_{t}(G)\leq a(G)+1$ holds for every nontrivial connected graph $G$. The conjecture has been proved for graphs with minimum degree at least $3$, trees, certain tree-like graphs, block graphs, and cactus graphs. In the main result of this paper it is proved that the conjecture holds for quasi-trees. The conjecture is verified also for some graph constructions including bijection graphs, Mycielskians, and the newly introduced universally-identifying graphs.

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arXiv2021

The domination game played on diameter 2 graphs

Let $γ_g(G)$ be the game domination number of a graph $G$. It is proved that if ${\rm diam}(G) = 2$, then $γ_g(G) \le \left\lceil \frac{n(G)}{2} \right\rceil- \left\lfloor \frac{n(G)}{11}\right\rfloor$. The bound is attained: if ${\rm diam}(G) = 2$ and $n(G) \le 10$, then $γ_g(G) = \left\lceil \frac{n(G)}{2} \right\rceil$ if and only if $G$ is one of seven sporadic graphs with $n(G)\le 6$ or the Petersen graph, and there are exactly ten graphs of diameter $2$ and order $11$ that attain the bound.

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arXiv2020

Comparing Wiener complexity with eccentric complexity

The transmission of a vertex $v$ of a graph $G$ is the sum of distances from $v$ to all the other vertices in $G$. The Wiener complexity of $G$ is the number of different transmissions of its vertices. Similarly, the eccentric complexity of $G$ is defined as the number of different eccentricities of its vertices. In this paper these two complexities are compared. The complexities are first studied on Cartesian product graphs. Transmission indivisible graphs and arithmetic transmission graphs are introduced to demonstrate sharpness of upper and lower bounds on the Wiener complexity, respectively. It is shown that for almost all graphs the Wiener complexity is not smaller than the eccentric complexity. This property is proved for trees, the equality holding precisely for center-regular trees. Several families of graphs in which the complexities are equal are constructed. Using the Cartesian product, it is proved that the eccentric complexity can be arbitrarily larger than the Wiener complexity. Additional infinite families of graphs with this property are constructed by amalgamating universally diametrical graphs with center-regular trees.

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arXiv2020

The general position number of the Cartesian product of two trees

The general position number of a connected graph is the cardinality of a largest set of vertices such that no three pairwise-distinct vertices from the set lie on a common shortest path. In this paper it is proved that the general position number is additive on the Cartesian product of two trees.

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arXiv2020

On Rall's $1/2$-conjecture on the domination game

The $1/2$-conjecture on the domination game asserts that if $G$ is a traceable graph, then the game domination number $γ_g(G)$ of $G$ is at most $\left\lceil \frac{n(G)}{2} \right\rceil$. A traceable graph is a $1/2$-graph if $γ_g(G) = \left\lceil \frac{n(G)}{2} \right\rceil$ holds. It is proved that the so-called hatted cycles are $1/2$-graphs and that unicyclic graphs fulfill the $1/2$-conjecture. Several additional families of graphs that support the conjecture are determined and computer experiments related to the conjecture described.

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arXiv2020

Constructing new families of transmission irregular graphs

The transmission of a vertex $v$ of a graph $G$ is the sum of distances from $v$ to all the other vertices in $G$. A graph is transmission irregular if all of its vertices have pairwise different transmissions. A starlike tree $T(k_1,\ldots,k_t)$ is a tree obtained by attaching to an isolated vertex $t$ pendant paths of lengths $k_1,\ldots,k_t$, respectively. It is proved that if a starlike tree $T(a,a+1,\ldots,a+k)$, $k\ge 2$, is of odd order, then it is transmission irregular. $T(1,2,\ldots,\ell)$, $\ell \ge 3$, is transmission irregular if and only if $\ell \notin \{r^2 + 1:\ r\ge 2\}$. Additional infinite families among the starlike trees and bi-starlike trees are determined. Transmission irregular unicyclic infinite families are also presented, in particular, the line graph of $T(a,a+1,a+2)$, $a\ge 2$, is transmission irregular if and only if $a$ is even.

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arXiv2019

Comparison of Wiener index and Zagreb eccentricity indices

The first and the second Zagreb eccentricity index of a graph $G$ are defined as $E_1(G)=\sum_{v\in V(G)}\varepsilon_{G}(v)^{2}$ and $E_2(G)=\sum_{uv\in E(G)}\varepsilon_{G}(u)\varepsilon_{G}(v)$, respectively, where $\varepsilon_G(v)$ is the eccentricity of a vertex $v$. In this paper the invariants $E_1$, $E_2$, and the Wiener index are compared on graphs with diameter $2$, on trees, on a newly introduced class of universally diametrical graphs, and on Cartesian product graphs. In particular, if the diameter of a tree $T$ is not too big, then $W(T) \ge E_2(T)$ holds, and if the diameter of $T$ is large, then $W(T) < E_1(T)$ holds.

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