arXiv subjects

Zhen-Mu Hong

Publications and source records attributed to Zhen-Mu Hong.

11 recordsLinked to original sources

arXiv2026

Characterization of the structure of $k$-edge-maximal graphs

Let $\kappa^{\prime}(G)$ be the edge-connectivity of the graph $G$. The \textit{strength} of $G$, denoted by $\overline{\kappa}^{\prime}(G)$, is the maximum edge-connectivity of its subgraphs. A simple graph $G$ is called $k$-\textit{edge-maximal} if $\overline{\kappa}^{\prime}(G) \leq k$ but for any edge $e$ not in $G$, $\overline{\kappa}^{\prime}(G+e) \geq k+1$. In this paper, we propose the concepts of kernel and closure of a graph and discuss the properties of closure. Utilizing these properties, we present the necessary and sufficient condition for a graph to be $k$-edge-maximal, which refines the results in [J. Graph Theory 14 (1990) 187--197], and prove that there exists a $k$-edge-maximal graph of order $n$ with $m$ edges if and only if $m=(n-1)k-\binom{k}{2}r$, for some integer $r$ with $1\leq r\leq \left\lfloor \frac{n}{k+2}\right\rfloor$. Furthermore, we characterize the structure of $k$-edge-maximal graphs with a given number of edges.

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arXiv2026

Sharp upper bounds on the $A_\alpha$-spectral radius of graphs

Let $G$ be a simple graph with degree diagonal matrix $D(G)$ and adjacency matrix $A(G)$. The signless Laplacian matrix of $G$ is defined as $Q(G)=D(G)+A(G)$. For a real number $\alpha \in [0, 1]$, Nikiforov (2017) proposed the $A_\alpha$-matrix of a graph $G$ as $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G)$. The $A_\alpha$-spectral radius of $G$, denoted by $\rho_\alpha(G)$, is the largest eigenvalue of $A_\alpha(G)$, where $\rho_0(G)=\rho(G)$ is the spectral radius of $A(G)$ and $2\rho_{\frac{1}{2}}(G)=q(G)$ is the spectral radius of $Q(G)$. Sun and Das (2020) proved that for any non-isolated vertex $v$ of degree $d_v$, $\rho^2(G)-\rho^2(G-v) \leq 2 d_v-1$, which confirmed the conjecture originally posed by Guo, Wang, and Li (2019). Recently, Liu and Ning (2026) provided a short and self-contained proof of this inequality. In this paper, we establish the corresponding result for $\rho_\alpha(G)$. As a corollary, for every $k\in [0,d_v+1]$, we have $$ \rho^2(G)- \rho^2(G-v) \leq 2d_v-1 +(k-2)\left(\frac{d_v}{\rho(G)}-1\right). $$ This inequality coincides with that of Sun and Das when $k=2$, and is strictly sharper than theirs whenever $k\neq 2$ and $d_v\neq \rho(G)$. We also give a short proof of the inequality $\rho_{\alpha}(G)-\rho_{\alpha}(G-v)\leq \alpha +\frac{(1-\alpha)^2d_v}{\rho_{\alpha}(G)-\alpha d_v}$, which is obtained by Wang and She (2022). Moreover, we obtain a unified generalization of Hong, Shu and Fang's inequality for $\rho(G)$ and Nikiforov's inequality for $q(G)$ in terms of $\rho_\alpha(G)$.

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arXiv2026

On the disjunctive domination numbers of the torus grid graphs

Let $\Gamma=(V,E)$ be a graph. The disjunctive domination number of $\Gamma$ is the minimum cardinality of a set $S\subseteq V$ such that every vertex not in $S$ is adjacent to a vertex of $S$, or has at least two vertices in $S$ at distance $2$ from it. In this paper, we give bounds for the disjunctive domination numbers of the torus grid graphs $C_m\Box C_n$, and determine the disjunctive domination numbers of $C_3\Box C_n$, $C_4\Box C_{n}$ and $C_8\Box C_{4n}$.

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arXiv2020

Induced subgraphs of product graphs and a generalization of Huang's theorem

Recently, Huang showed that every $(2^{n-1}+1)$-vertex induced subgraph of the $n$-dimensional hypercube has maximum degree at least $\sqrt{n}$ in [Annals of Mathematics, 190 (2019), 949--955]. In this paper, we discuss the induced subgraphs of Cartesian product graphs and semi-strong product graphs to generalize Huang's result. Let $Γ_1$ be a connected signed bipartite graph of order $n$ and $Γ_2$ be a connected signed graph of order $m$. By defining two kinds of signed product of $Γ_1$ and $Γ_2$, denoted by $Γ_1\widetilde{\Box}Γ_2$ and $Γ_1\widetilde{\bowtie} Γ_2$, we show that if $Γ_1$ and $Γ_2$ have exactly two distinct adjacency eigenvalues $\pmθ_1$ and $\pmθ_2$ respectively, then every $(\frac{1}{2}mn+1)$-vertex induced subgraph of $Γ_1\widetilde{\Box}Γ_2$ (resp. $Γ_1\widetilde{\bowtie} Γ_2$) has maximum degree at least $\sqrt{θ_1^2+θ_2^2}$ (resp. $\sqrt{(θ_1^2+1)θ_2^2}$). Moreover, we discuss the eigenvalues of $Γ_1\widetilde{\Box} Γ_2$ and $Γ_1\widetilde{\bowtie} Γ_2$ and obtain a sufficient and necessary condition such that the spectrum of $Γ_1\widetilde{\Box}Γ_2$ and $Γ_1\widetilde{\bowtie}Γ_2$ are symmetric, from which we obtain more general results on maximum degree of the induced subgraphs.

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arXiv2020

Connectivity and eigenvalues of graphs with given girth or clique number

Let $κ'(G)$, $κ(G)$, $μ_{n-1}(G)$ and $μ_1(G)$ denote the edge-connectivity, vertex-connectivity, the algebraic connectivity and the Laplacian spectral radius of $G$, respectively. In this paper, we prove that for integers $k\geq 2$ and $r\geq 2$, and any simple graph $G$ of order $n$ with minimum degree $δ\geq k$, girth $g\geq 3$ and clique number $ω(G)\leq r$, the edge-connectivity $κ'(G)\geq k$ if $μ_{n-1}(G) \geq \frac{(k-1)n}{N(δ,g)(n-N(δ,g))}$ or if $μ_{n-1}(G) \geq \frac{(k-1)n}{φ(δ,r)(n-φ(δ,r))}$, where $N(δ,g)$ is the Moore bound on the smallest possible number of vertices such that there exists a $δ$-regular simple graph with girth $g$, and $φ(δ,r) = \max\{δ+1,\lfloor\frac{rδ}{r-1}\rfloor\}$. Analogue results involving $μ_{n-1}(G)$ and $\frac{μ_1(G)}{μ_{n-1}(G)}$ to characterize vertex-connectivity of graphs with fixed girth and clique number are also presented. Former results in [Linear Algebra Appl. 439 (2013) 3777--3784], [Linear Algebra Appl. 578 (2019) 411--424], [Linear Algebra Appl. 579 (2019) 72--88], [Appl. Math. Comput. 344-345 (2019) 141--149] and [Electronic J. Linear Algebra 34 (2018) 428--443] are improved or extended.

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arXiv2019

A note on the optimal rubbling in ladders and prisms

A pebbling move on a graph G consists of the removal of two pebbles from one vertex and the placement of one pebble on an adjacent vertex. Rubbling is a version of pebbling where an additional move is allowed, which is also called the strict rubbling move. In this new move, one pebble each is removed from u and v adjacent to a vertex w, and one pebble is added on w. The optimal rubbling number of a graph G is the smallest number m, such that one pebble can be moved to every given vertex from some pebble distribution of m pebbles by a sequence of rubbling moves. In this paper, we give short proofs to determine the rubbling number of cycles and the optimal rubbling number of paths, cycles, ladders, prisms and Mobius-ladders.

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arXiv2019

Generalization of the cover pebbling number on trees

A pebbling move on a graph consists of taking two pebbles off from one vertex and add one pebble on an adjacent vertex, the $t$-pebbling number of a graph $G$ is the minimum number of pebbles so that we can move $t$ pebbles on any vertex on $G$ regardless the original distribution of pebbles. Let $ω$ be a positive function on $V(G)$, the $ω$-cover pebbling number of a graph $G$ is the minimum number of pebbles so that we can reach a distribution with at least $ω(v)$ pebbles on $v$ for all $v\in V(G)$. In this paper, we give the $ω$-cover pebbling number of trees for nonnegative function $ω$, which generalized the $t$-pebbling number and the traditional weighted cover pebbling number of trees.

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arXiv2017

Sufficient conditions for graphs to be $k$-connected, maximally connected and super-connected

Let $G$ be a connected graph with minimum degree $δ(G)$ and vertex-connectivity $κ(G)$. The graph $G$ is $k$-connected if $κ(G)\geq k$, maximally connected if $κ(G) = δ(G)$, and super-connected (or super-$κ$) if every minimum vertex-cut isolates a vertex of minimum degree. In this paper, we show that a connected graph or a connected triangle-free graph is $k$-connected, maximally connected or super-connected if the number of edges or the spectral radius is large enough.

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arXiv2017

Pebbling on Jahangir graphs

The pebbling number of a graph $G$, $f(G)$, is the least $p$ such that, however $p$ pebbles are placed on the vertices of $G$, we can move a pebble to any vertex by a sequence of moves, each move taking two pebbles off one vertex and placing one on an adjacent vertex. In this paper, we will show the pebbling number of Jahangir graphs $J_{n,m}$ with $n$ even, $m\geq8$.

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arXiv2016

On restricted edge-connectivity of replacement product graphs

This paper considers the edge-connectivity and restricted edge-connectivity of replacement product graphs, gives some bounds on edge-connectivity and restricted edge-connectivity of replacement product graphs and determines the exact values for some special graphs. In particular, the authors further confirm that under certain conditions, the replacement product of two Cayley graphs is also a Cayley graph, and give a necessary and sufficient condition for such Cayley graphs to have maximum restricted edge-connectivity. Based on these results, the authors construct a Cayley graph with degree $d$ whose restricted edge-connectivity is equal to $d+s$ for given odd integer $d$ and integer $s$ with $d \geqslant 5$ and $1\leqslant s\leqslant d-3$, which answers a problem proposed ten years ago.

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arXiv2013

Vulnerability of super edge-connected graphs

A subset $F$ of edges in a connected graph $G$ is a $h$-extra edge-cut if $G-F$ is disconnected and every component has more than $h$ vertices. The $h$-extra edge-connectivity $\la^{(h)}(G)$ of $G$ is defined as the minimum cardinality over all $h$-extra edge-cuts of $G$. A graph $G$, if $\la^{(h)}(G)$ exists, is super-$\la^{(h)}$ if every minimum $h$-extra edge-cut of $G$ isolates at least one connected subgraph of order $h+1$. The persistence $ρ^{(h)}(G)$ of a super-$\la^{(h)}$ graph $G$ is the maximum integer $m$ for which $G-F$ is still super-$\la^{(h)}$ for any set $F\subseteq E(G)$ with $|F|\leqslant m$. Hong {\it et al.} [Discrete Appl. Math. 160 (2012), 579-587] showed that $\min\{\la^{(1)}(G)-δ(G)-1,δ(G)-1\}\leqslant ρ^{(0)}(G)\leqslant δ(G)-1$, where $δ(G)$ is the minimum vertex-degree of $G$. This paper shows that $\min\{\la^{(2)}(G)-ξ(G)-1,δ(G)-1\}\leqslant ρ^{(1)}(G)\leqslant δ(G)-1$, where $ξ(G)$ is the minimum edge-degree of $G$. In particular, for a $k$-regular super-$\la'$ graph $G$, $ρ^{(1)}(G)=k-1$ if $\la^{(2)}(G)$ does not exist or $G$ is super-$\la^{(2)}$ and triangle-free, from which the exact values of $ρ^{(1)}(G)$ are determined for some well-known networks.

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