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Leyou Xu

Publications and source records attributed to Leyou Xu.

At least 19 recordsLinked to original sources
arXiv2026

Proving a conjecture concerning chromatic number, size and least eigenvalue

Let $G$ be a simple nonempty graph with size $m$, chromatic number $\chi$, and least eigenvalue $\lambda$. We prove that \[ \chi(\chi-1) \le (m+1-\lambda^2)+\sqrt{(m+1-\lambda^2)^2-4(\lambda^2-1)(\lambda^2-m)} \] with equality if and only if $G$ is either a complete graph or a complete bipartite graph, with possibly isolated vertices. The inequality was conjectured recently by Tang and Elphick in [Electron. J. Combin. 33 (2026), \#P2.65].

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arXiv2026

The number of cycles of a given length in dense hamiltonian graphs: proving Hilton's conjecture

A classical theorem of Sheehan in 1977 states that every hamiltonian graph $G$ of order $n$ satisfying $e(G)>\left\lfloor \frac{n^2}{4}\right\rfloor+1$ contains at least two cycles of every length $\ell$, $3\le \ell\le n$. In the same paper, Sheehan recorded a conjecture of Hilton, which strengthens this conclusion by asserting that such a graph contains at least $n-\ell+2$ cycles of length $\ell$ for each $3\le \ell\le n$. We prove Hilton's conjecture for all hamiltonian graphs of order at least $440$.

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arXiv2026

On $k$-connected vertex-pancyclic graphs without pancyclic edges

An edge of a graph of order $n$ is pancyclic if it lies in a cycle of every length $3,\ldots,n$. A graph of order $n$ is vertex-pancyclic if every vertex lies in a cycle of every length $3,\ldots,n$. Recently, Li and Zhan proved that every $2$-connected $[4,2]$-graph of order at least seven contains a pancyclic edge. Zhan asked whether there exists a positive integer $k$ such that every $k$-connected vertex-pancyclic graph contains a pancyclic edge. We answer this question by showing that for every positive integer $k$, there is a $k$-connected vertex-pancyclic graph containing no pancyclic edge.

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arXiv2026

Extremal $Q$-index problem in outerplanar graphs

Outerplanar Tur\'an problem has received considerable attention recently. We study the spectral version via $Q$-index. We determine the unique graph that maximizes the $Q$-index among all $n$-vertex connected outerplanar graphs which are respectively forbidden to contain: (i) a fixed cycle; and (ii) the disjoint union of paths of a given order.

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arXiv2025

On the multiplicity of matching polynomial roots and $\theta$-critical graphs

The matching polynomial of a graph encodes rich combinatorial information through its roots. We determine the maximum multiplicity of a non-zero matching polynomial root and characterize all graphs attaining the bound. We also generalize the result to any fixed $\theta$, where the graphs attaining the bound are related to $\theta$-critical graphs. Inspired by these graphs, we give a constructive answer to Godsil's question. Finally, we show the existence of $1$-critical tree of order $n$ for all $n\ge 9$ and $1$-critical graph of order $n$ for all $n\ge 5$, and describe a method to construct $1$-critical graphs from existing ones.

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arXiv2025

Proof of a conjecture on eigenvalues of transposition graph

The transposition graph $Cay(S_n,T_n)$ is the Cayley graph on the symmetric group $S_n$ generated by the set $T_n$ of all transpositions. In this paper, we show that each integer in the interval $\left[-{\lfloor(2n+1)/3 \rfloor\choose 2}, {\lfloor(2n+1)/3 \rfloor\choose 2}\right]$ is an eigenvalue of $Cay(S_n,T_n)$. This proves a recent conjecture by Kravchuk \cite{Kravchuk}.

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arXiv2025

Girth and Laplacian eigenvalue distribution

Let $G$ be a connected graph of order $n$ with girth $g$. For $k=1,\dots,\min\{g-1, n-g\}$, let $n(G,k)$ be the number of Laplacian eigenvalues (counting multiplicities) of $G$ that fall inside the interval $[n-g-k+4,n]$. We prove that if $g\ge 4$, then \[ n(G,k)\le n-g. \] Those graphs achieving the bound for $k=1,2$ are determined. We also determine the graphs $G$ with $g=3$ such that $n(G,k)=n-1, n-2, n-3$.

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arXiv2025

Cycles and paths through specified vertices in graphs with a given clique number

B. Bollob\'{a}s and G. Brightwell and independently R. Shi proved the existence of a cycle through all vertices whose degrees at least $\frac{n}{2}$ in any $2$-connected graph of order $n$. Motivated by this result, we prove the existence of a cycle through all vertices whose degrees at least $n-\omega$ in any $2$-connected graph $G$ of order $n$ with clique number $\omega$ unless $G$ is a specific graph. Moreover, we show that for any pair of vertices whose degrees are at least $n-\omega+1$ in a graph $G$ of order $n$ with clique number $\omega$, there exists a path joining them which contains all vertices of degree at least $n-\omega+1$ unless $G$ belongs to certain graph classes. In doing so, we prove the existence of a $(u,v)$-path through all vertices whose degrees at least $\frac{n+1}{2}$ in any graph of order $n$, where $u,v$ are two distinct vertices of degree at least $\frac{n+1}{2}$.

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arXiv2024

Spectral conditions for the existence of chorded cycles in graphs with fixed size

A chorded cycle is a cycle with at least one chord. Gould asked in [Graphs Comb. 38 (2022) 189] the question: What spectral conditions imply a graph contains a chorded cycle? For a graph with fixed size, extremal spectral conditions are given to ensure that a graph contains a chorded cycle and a $(2k-3)$-chorded $(2k+1)$-cycle for $k\ge 2$, respectively, via spectral radius.

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arXiv2024

Distribution of signless Laplacian eigenvalues and graph invariants

For a simple graph on $n$ vertices, any of its signless Laplacian eigenvalues is in the interval $[0, 2n-2]$. In this paper, we give relationships between the number of signless Laplacian eigenvalues in specific intervals in $[0, 2n-2]$ and graph invariants including matching number and diameter.

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arXiv2024

Diameter vs Laplacian eigenvalue distribution

Let $G$ be a simple graph of order $n$. It is known that any Laplacian eigenvalue of $G$ belongs to the interval $[0,n]$. For an interval $I\subseteq [0, n]$, denote by $m_GI$ the number of Laplacian eigenvalues of $G$ in $I$, counted with multiplicity. When $G$ is connected, known results on the Laplacian eigenvalue distribution related to the diameter $d$ of $G$ include: $m_G[n-d+2,n]\le n-d$ if $2\le d\le n-3$ and $m_G[n-d+1,n]\le n-d+1$ if $1\le d\le n-3$. In this paper, we show that $m_G[n-d,n]\le n-d+2$ if $2\le d\le n-4$, and $m_G[n-2d+4,n]\le n-2$ if $2\le d\le \lfloor\frac{n}{2} \rfloor$.

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arXiv2023

Link residual closeness of graphs with fixed parameters

Link residual closeness is a newly proposed measure for network vulnerability. In this model, vertices are perfectly reliable and the links fail independently of each other. It measures the vulnerability even when the removal of links does not disconnect the graph. In this paper, we characterize those graphs that maximize the link residual closeness over the connected graphs with fixed order and one parameters such as connectivity, edge connectivity, bipartiteness, independence number, matching number, chromatic number, number of vertices and number of cut edges.

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arXiv2023

The Alon-Tarsi number of $K_{3,3}$-minor-free graphs

The well known Wagner's theorem states that a graph is a planar graph if and only if it is $K_5$-minor-free and $K_{3,3}$-minor-free. Denote by $AT(G)$ the Alon-Tarsi number of a graph $G$. We show that for any $K_{3,3}$-minor-free graph $G$, $AT(G)\le 5$, there exists a matching $M$ and a forest $F$ such that $AT(G-M)\le 4$ and $AT(G-E(F))\le 3$, extending the result on the Alon-Tarsi number of $K_5$-minor-free graphs due to Abe, Kim and Ozeki.

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arXiv2023

Toughness and existence of $2$-factors

A graph is $t$-tough if the deletion of any set of, say, $m$ vertices from the graph leaves a graph with at most $\frac{m}{t}$ components. In 1973, Chvátal suggested the problem of relating toughness to factors in graphs. In 1985, Enomoto et al. showed that each $2$-tough graph with at least three vertices has a $2$-factor, but for any $ε>0$, there exists a $(2-ε)$-tough graph on at least $3$ vertices having no $2$-factor. In recent years, the study of sufficient conditions for graphs with toughness less than $2$ having a $2$-factor has received a paramount interest. In this paper, we give new tight sufficient conditions for a $t$-tough graph having a $2$-factor when $1\le t<2$ by involving independence number, minimum degree, connectivity and forbidden forests.

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arXiv2023

Laplacain eigenvalue distribution and diameter of graphs

Let $G$ be a connected graph on $n$ vertices with diameter $d$. It is known that if $2\le d\le n-2$, there are at most $n-d$ Laplacian eigenvalues in the interval $[n-d+2, n]$. In this paper, we show that if $1\le d\le n-3$, there are at most $n-d+1$ Laplacian eigenvalues in the interval $[n-d+1, n]$. Moreover, we try to identify the connected graphs on $n$ vertices with diameter $d$, where $2\le d\le n-3$, such that there are at most $n-d$ Laplacian eigenvalues in the interval $[n-d+1, n]$.

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arXiv2023

Proof of a conjecture on distribution of Laplacian eigenvalues and diameter, and beyond

Ahanjideh, Akbari, Fakharan and Trevisan proposed a conjecture in [Linear Algebra Appl. 632 (2022) 1--14] on the distribution of the Laplacian eigenvalues of graphs: for any connected graph of order $n$ with diameter $d\ge 2$ that is not a path, the number of Laplacian eigenvalues in the interval $[n-d+2,n]$ is at most $n-d$. We show that the conjecture is true, and give a complete characterization of graphs for which the conjectured bound is attained. This establishes an interesting relation between the spectral and classical parameters.

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