arXiv subjects

Sizhong Zhou

Publications and source records attributed to Sizhong Zhou.

At least 19 recordsLinked to original sources

arXiv2026

Distance spectral radius and perfect matchings in graphs with given fractional property

A matching in a graph $G$ is a set of independent edges in $G$. A perfect matching in a graph $G$ is a matching which saturates all the vertices of $G$. A fractional perfect matching in a graph $G$ is a function $h:E(G)\rightarrow [0,1]$ such that $\sum\limits_{e\in E_G(v)}h(e)=1$ for every $v\in V(G)$, where $E_G(v)$ is the set of edges incident to $v$ in $G$. Clearly, the existence of a fractional perfect matching in a graph is a necessary condition for the graph to possess a perfect matching. Let $G$ be a $k$-connected graph of even order $n$ with a fractional perfect matching, where $k$ is a positive integer. We denote by $μ(G)$ the distance spectral radius of $G$. In this paper, we prove that if $n\geq8k+6$ and $μ(G)\leqμ(K_k\vee(kK_1\cup K_3\cup K_{n-2k-3}))$, then $G$ contains a perfect matching unless $G=K_k\vee(kK_1\cup K_3\cup K_{n-2k-3})$.

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arXiv2026

The existence of odd-even factors in 1-binding graphs

Let $G$ be a graph. The binding number of $G$, denoted by $\mbox{bind}(G)$, is defined as $$ \mbox{bind}(G)=\min\left\{\frac{|N_G(S)|}{|S|}:\emptyset\neq S\subseteq V(G) \ \mbox{and} \ N_G(S)\neq V(G)\right\}. $$ If $\mbox{bind}(G)\geq r$, then $G$ is called $r$-binding, where $r$ is a positive real number. The adjacency matrix of $G$ is denoted by $A(G)$. The largest eigenvalue of $A(G)$, denoted by $ρ(G)$, is said to be the spectral radius of $G$. A spanning subgraph $F$ of $G$ is called an odd-even factor $F=F_W$ if $d_F(u)\in\{1,3,\ldots,k\}$ for every $u\in W$ and $d_F(v)\in\{0,2,\ldots,k+1\}$ for every $v\in V(G)-W$, where $k$ is a positive odd integer and $W$ is any set of even number of vertices of $G$. In this paper, we propose a tight sufficient condition based on the spectral radius to guarantee that a connected 1-binding graph $G$ contains an odd-even factor $F=F_W$ such that $d_F(u)\in\{1,3,\ldots,k\} \ \mbox{for all} \ u\in W$ and $d_F(v)\in\{0,2,\ldots,k+1\} \ \mbox{for all} \ v\in V(G)-W$.

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arXiv2026

Sufficient conditions for a special factor in a graph with minimum degree

Let $G$ be a graph. The size and the signless Laplacian spectral radius of $G$ are denoted by $e(G)$ and $q(G)$, respectively. A spanning subgraph $F$ of $G$ is called an $H_b$-factor of $G$ if $d_F(v)\in\{1,3,5,\ldots,b-1,b\}$ for every $v\in V(G)$, where $b\geq2$ is an even integer. Lu and Wang obtained a sufficient condition according to the number of odd components in $G-S$ for a connected graph $G$ of even order to have an $H_b$-factor, where $S$ is a subset of $V(G)$ [H. Lu, D. Wang, On Cui-Kano's characterization problem on graph factors, J. Graph Theory 74 (2013) 335--343]. In this paper, motivated by Lu and Wang's above result, we establish a lower bound for the size in an $n$-vertex connected graph $G$ with given minimum degree to guarantee that $G$ has an $H_b$-factor. Further, we show a lower bound for the signless Laplacian spectral radius in an $n$-vertex 2-connected graph $G$ with given minimum degree to ensure that $G$ has an $H_b$-factor.

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arXiv2026

Spectral radius and perfect k-matchings in t-connected graphs

A $k$-matching of a graph $G$ is a function $f:E(G)\rightarrow\{0,1,2,\ldots,k\}$ with $\sum\limits_{e\in E_G(v)}f(e)\leq k$ for each vertex $v$ of $G$, where $E_G(v)$ is the set of edges incident with $v$ in $G$. A perfect $k$-matching of a graph $G$ is a $k$-matching $f$ satisfying $\sum\limits_{e\in E_G(v)}f(e)=k$ for any vertex $v$ of $G$. A fractional perfect matching of a graph $G$ is a function $f:E(G)\rightarrow [0,1]$ satisfying $\sum\limits_{e\in E_G(v)}f(e)=1$ for any $v\in V(G)$. We denote by $ρ(G)$ the spectral radius of $G$. In this paper, we put forward a tight spectral radius condition for a $t$-connected graph to possess a perfect $k$-matching and a tight spectral radius condition for the existence of a perfect $k$-matching in a $t$-connected graph with a fractional perfect matching.

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arXiv2026

Perfect matchings and $A_α$-spectral radius in 1-binding graphs

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. For $α\in[0,1)$, we use $A_α(G)$ and $ρ_α(G)$ to denote the $A_α$-matrix and the $A_α$-spectral radius of $G$, respectively. The binding number $\mbox{bind}(G)$ of $G$ is defined by $\mbox{bind}(G)=\min\left\{\frac{|N_G(X)|}{|X|}:\emptyset\neq X\subseteq V(G),N_G(X)\neq V(G)\right\}$. If $\mbox{bind}(G)\geq1$, then $G$ is called 1-binding. A perfect matching in $G$ is a set of nonadjacent edges covering every vertex of $G$. Tutte proved that a graph $G$ of even order has a perfect matching if and only if $o(G-S)\leq|S|$ holds for every $S\subseteq V(G)$ [W. Tutte, The factorization of linear graphs, J. Lond. Math. Soc. 22 (1947) 107--111]. In this paper, we use Tutte's result to prove that a connected 1-binding graph $G$ of even order $n$ with $n\geq n(α)$ has a perfect matching unless $G=K_1\vee(K_{n-5}\cup K_3\cup K_1)$ if $ρ_α(G)\geqρ_α(K_1\vee(K_{n-5}\cup K_3\cup K_1))$, where $n(α)$ is defined as follows: $n(α)=\max\{18,\frac{2+8α}{1-2α}\}$ if $α\in[0,\frac{1}{2})$, and $n(α)=18$ if $α=\frac{1}{2}$.

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arXiv2026

Spectral radii and star-factors with large components

Let $G$ be a connected graph with $n$ vertices. The isolated toughness of $G$, denoted by $I(G)$, is defined by $I(G)=\min\left\{\frac{|S|}{i(G-S)}:S\subseteq V(G) \ \mbox{and} \ i(G-S)\geq2\right\}$ if $G$ is not complete, or $I(G)=+\infty$ if $G$ is complete. A graph $G$ is called isolated $r$-tough if $I(G)\geq r$. A spanning subgraph $H$ of $G$ is called a $\{K_{1,j}:m\leq j\leq2m\}$-factor of $G$ if every component of $H$ is isomorphic to an element of $\{K_{1,j}:m\leq j\leq2m\}$. Let $ρ(G)$, $q(G)$ and $μ(G)$ denote the adjacency spectral radius, the signless Laplacian spectral radius and the distance spectral radius of $G$, respectively. Let $m$ and $b$ be two positive integers with $m\geq2$. In this paper, we first establish a lower bounds on the adjacency spectral radius of a connected isolated $\frac{mb-1}{b}$-tough graph $G$ to guarantees that $G$ contains a $\{K_{1,j}:m\leq j\leq2m\}$-factor. Second, we establish a lower bounds on the signless Laplacian spectral radius of a connected isolated $\frac{mb-1}{b}$-tough graph $G$ to ensures that $G$ contains a $\{K_{1,j}:m\leq j\leq2m\}$-factor. Finally, we create an upper bounds on the distance spectral radius of a connected isolated $\frac{mb-1}{b}$-tough graph $G$ with a $\{K_{1,j}:m\leq j\leq2m\}$-factor. Furthermore, we construct some extremal graphs to claim that all the bounds obtained in this paper are sharp.

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arXiv2026

Toughness and Aα-spectral radius in graphs

Let $α\in[0,1)$, and let $G$ be a connected graph of order $n$ with $n\geq f(α)$, where $f(α)=6$ for $α\in[0,\frac{2}{3}]$ and $f(α)=\frac{4}{1-α}$ for $α\in(\frac{2}{3},1)$. A graph $G$ is said to be $t$-tough if $|S|\geq tc(G-S)$ for each subset $S$ of $V(G)$ with $c(G-S)\geq2$, where $c(G-S)$ is the number of connected components in $G-S$. The $A_α$-spectral radius of $G$ is denoted by $ρ_α(G)$. In this paper, it is verified that $G$ is a 1-tough graph unless $G=K_1\vee(K_{n-2}\cup K_1)$ if $ρ_α(G)\geqρ_α(K_1\vee(K_{n-2}\cup K_1))$, where $ρ_α(K_1\vee(K_{n-2}\cup K_1))$ equals the largest root of $x^{3}-((α+1)n+α-3)x^{2}+(αn^{2}+(α^{2}-α-1)n-2α+1)x-α^{2}n^{2}+(3α^{2}-α+1)n-4α^{2}+5α-3=0$. Further, we present an $A_α$-spectral radius condition for a graph to be a $t$-tough graph.

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arXiv2026

A result on spanning trees with bounded total excess

Let $G$ be a connected graph and $T$ a spanning tree of $G$. Let $ρ(G)$ denote the adjacency spectral radius of $G$. The $k$-excess of a vertex $v$ in $T$ is defined as $\max\{0,d_T(v)-k\}$. The total $k$-excess $\mbox{te}(T,k)$ is defined by $\mbox{te}(T,k)=\sum\limits_{v\in V(T)}{\max\{0,d_T(v)-k\}}$. A tree $T$ is said to be a $k$-tree if $d_T(v)\leq k$ for any $v\in V(T)$, that is to say, the maximum degree of a $k$-tree is at most $k$. In fact, $T$ is a spanning $k$-tree if and only if $\mbox{te}(T,k)=0$. This paper studies a generalization of spanning $k$-trees using a concept called total $k$-excess and proposes a lower bound for $ρ(G)$ in a connected graph $G$ to ensure that $G$ contains a spanning tree $T$ with $\mbox{te}(T,k)\leq b$, where $k$ and $b$ are two nonnegative integers with $k\geq\max\{5,b+3\}$ and $(b,k)\neq(2,5)$.

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arXiv2025

Generalized toughness and Q-index in a graph

Let $G$ be a graph. We denote by $c(G)$, $α(G)$ and $q(G)$ the number of components, the independence number and the signless Laplacian spectral radius ($Q$-index for short) of $G$, respectively. The toughness of $G$ is defined by $t(G)=\min\left\{\frac{|S|}{c(G-S)}:S\subseteq V(G), c(G-S)\geq2\right\}$ for $G\neq K_n$ and $t(G)=+\infty$ for $G=K_n$. Chen, Gu and Lin [Generalized toughness and spectral radius of graphs, Discrete Math. 349 (2026) 114776] generalized this notion and defined the $l$-toughness $t_l(G)$ of a graph $G$ as $t_l(G)=\min\left\{\frac{|S|}{c(G-S)}:S\subset V(G), c(G-S)\geq l\right\}$ if $2\leq l\leqα(G)$, and $t_l(G)=+\infty$ if $l>α(G)$. If $t_l(G)\geq t$, then $G$ is said to be $(t,l)$-tough. In this paper, we put forward $Q$-index conditions for a graph to be $(b,l)$-tough and $(\frac{1}{b},l)$-tough, respectively.

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arXiv2025

Sufficient conditions for even factors in graphs

Let $G$ be a graph. We denote by $e(G)$ and $ρ(G)$ the size and the spectral radius of $G$. A spanning subgraph $F$ of $G$ is called an even factor of $G$ if $d_F(v)\in\{2,4,6,\ldots\}$ for every $v\in V(G)$. Yan and Kano provided a sufficient condition using the number of odd components in $G-S$ for a graph $G$ of even order to contain an even factor, where $S$ is a vertex subset of $G$ [Z. Yan, M. Kano, Strong Tutte type conditions and factors of graphs, Discuss. Math. Graph Theory 40 (2020) 1057--1065]. In this paper, motivated by Yan and Kano's above result, we present some tight sufficient conditions to guarantee that a connected graph $G$ with the minimum degree $δ$ contains an even factor with respect to its size and spectral radius.

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arXiv2025

Adjacency spectral radius and H-factors in 1-binding graphs

Let $G$ be a graph, and let $H:V(G)\longrightarrow\{\{1\},\{0,2\}\}$ be a set-valued function. Hence, $H(v)$ equals $\{1\}$ or $\{0,2\}$ for any $v\in V(G)$. We let $$ H^{-1}(1)=\{v: v\in V(G) \ \mbox{and} \ H(v)=1\}. $$ An $H$-factor of $G$ is a spanning subgraph $F$ of $G$ such that $d_F(v)\in H(v)$ for each $v\in V(G)$. Lu and Kano showed a characterization for the existence of an $H$-factor in a graph [Characterization of 1-tough graphs using factors, Discrete Math. 343 (2020) 111901]. Let $A(G)$ and $ρ(G)$ denote the adjacency matrix and the adjacency spectral radius of $G$, respectively. By using Lu and Kano's result, we pose a sufficient condition with respect to the adjacency spectral radius to guarantee the existence of an $H$-factor in a 1-binding graph. In this paper, we prove that if a connected 1-binding graph $G$ of order $n\geq11$ satisfies $ρ(G)\geqρ(K_1\vee(K_{n-4}\cup K_2\cup K_1))$, then $G$ has an $H$-factor for each $H:V(G)\longrightarrow\{\{1\},\{0,2\}\}$ with $H^{-1}(1)$ even, unless $G=K_1\vee(K_{n-4}\cup K_2\cup K_1)$.

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arXiv2025

Spanning subgraphs and spectral radius in graphs

A spanning tree $T$ of a connected graph $G$ is a subgraph of $G$ that is a tree covers all vertices of $G$. The leaf distance of $T$ is defined as the minimum of distances between any two leaves of $T$. A fractional matching of a graph $G$ is a function $h$ assigning every edge a real number in $[0,1]$ so that $\sum\limits_{e\in E_G(v)}{h(e)}\leq1$ for any $v\in V(G)$, where $E_G(v)$ denotes the set of edges incident with $v$ in $G$. A fractional matching of $G$ is called a fractional perfect matching if $\sum\limits_{e\in E_G(v)}{h(e)}=1$ for any $v\in V(G)$. A graph $G$ with at least $2k+2$ vertices is said to be fractional $k$-extendable if every $k$-matching $M$ in $G$ is included in a fractional perfect matching $h$ of $G$ such that $h(e)=1$ for any $e\in M$. This paper considers a lower bound on the spectral radius of $G$ to guarantee that $G$ has a spanning tree with leaf distance at least $d$. At the same time, we obtain a lower bound on the spectral radius of $G$ to ensure that $G$ is fractional $k$-extendable.

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arXiv2025

Spanning k-trees, odd [1,b]-factors and spectral radius in binding graphs

The binding number of a graph $G$, written as $\mbox{bind}(G)$, is defined by $$ \mbox{bind}(G)=\min\left\{\frac{|N_G(X)|}{|X|}:\emptyset\neq X\subseteq V(G),N_G(X)\neq V(G)\right\}. $$ A graph $G$ is called $r$-binding if $\mbox{bind}(G)\geq r$. An odd $[1,b]$-factor of a graph $G$ is a spanning subgraph $F$ with $d_F(v)\in\{1,3,\ldots,b\}$ for all $v\in V(G)$, where $b\geq1$ is an odd integer. A spanning $k$-tree of a connected graph $G$ is a spanning tree $T$ with $d_T(v)\leq k$ for every $v\in V(G)$. In this paper, we first show a tight sufficient condition with respect to the adjacency spectral radius for connected $\frac{1}{b}$-binding graphs to have odd $[1,b]$-factors, which generalizes Fan and Lin's previous result [D. Fan, H. Lin, Binding number, $k$-factor and spectral radius of graphs, Electron. J. Combin. 31(1) (2024) \#P1.30] and partly improves Fan, Liu and Ao's previous result [A. Fan, R. Liu, G. Ao, Spectral radius, odd $[1,b]$-factor and spanning $k$-tree of 1-binding graphs, Linear Algebra Appl. 705 (2025) 1--16]. Then we put forward a tight sufficient condition via the adjacency spectral radius for connected $\frac{1}{k-2}$-binding graphs to have spanning $k$-trees, which partly improves Fan, Liu and Ao's previous result [A. Fan, R. Liu, G. Ao, Spectral radius, odd $[1,b]$-factor and spanning $k$-tree of 1-binding graphs, Linear Algebra Appl. 705 (2025) 1--16].

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arXiv2025

Sufficient conditions for a graph with minimum degree to be k-critical with respect to [1,b]-odd factor

A spanning subgraph $F$ of a graph $G$ is called a $[1,b]$-odd factor if $b\equiv1$ (mod 2) and $d_F(v)\in\{1,3,\ldots,b\}$ for every $v\in V(G)$. A graph $G$ of order $n\geq k+2$ is $k$-critical with respect to $[1,b]$-odd factor if for any $X\subseteq V(G)$ with $|X|=k$, $G-X$ has a $[1,b]$-odd factor. In this paper, we provide a size and spectral radius conditions for a graph with minimum degree to be $k$-critical with respect to $[1,b]$-odd factor, respectively.

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arXiv2025

An Aα-spectral radius for the existence of {P3, P4, P5}-factors in graphs

Let $G$ be a connected graph of order $n$ with $n\geq25$. A $\{P_3,P_4,P_5\}$-factor is a spanning subgraph $H$ of $G$ such that every component of $H$ is isomorphic to an element of $\{P_3,P_4,P_5\}$. Nikiforov introduced the $A_α$-matrix of $G$ as $A_α(G)=αD(G)+(1-α)A(G)$ [V. Nikiforov, Merging the $A$- and $Q$-spectral theories, Appl. Anal. Discrete Math. 11 (2017) 81--107], where $α\in[0,1]$, $D(G)$ denotes the diagonal matrix of vertex degrees of $G$ and $A(G)$ denotes the adjacency matrix of $G$. The largest eigenvalue of $A_α(G)$, denoted by $λ_α(G)$, is called the $A_α$-spectral radius of $G$. In this paper, it is proved that $G$ has a $\{P_3,P_4,P_5\}$-factor unless $G=K_1\vee(K_{n-2}\cup K_1)$ if $λ_α(G)\geqλ_α(K_1\vee(K_{n-2}\cup K_1))$, where $α$ be a real number with $0\leqα<\frac{2}{3}$.

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arXiv2024

A spectral condition for spanning trees with restricted degrees in bipartite graphs

Let $G$ be a graph and $T$ be a spanning tree of $G$. We use $Q(G)=D(G)+A(G)$ to denote the signless Laplacian matrix of $G$, where $D(G)$ is the diagonal degree matrix of $G$ and $A(G)$ is the adjacency matrix of $G$. The signless Laplacian spectral radius of $G$ is denoted by $q(G)$. A necessary and sufficient condition for a connected bipartite graph $G$ with bipartition $(A,B)$ to have a spanning tree $T$ with $d_T(v)\geq k$ for any $v\in A$ was independently obtained by Frank and Gyárfás (A. Frank, E. Gyárfás, How to orient the edges of a graph?, Colloq. Math. Soc. Janos Bolyai 18 (1976) 353--364), Kaneko and Yoshimoto (A. Kaneko, K. Yoshimoto, On spanning trees with restricted degrees, Inform. Process. Lett. 73 (2000) 163--165). Based on the above result, we establish a lower bound on the signless Laplacian spectral radius $q(G)$ of a connected bipartite graph $G$ with bipartition $(A,B)$, in which the bound guarantees that $G$ has a spanning tree $T$ with $d_T(v)\geq k$ for any $v\in A$.

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arXiv2024

A spectral condition for a graph having a strong parity factor

A graph $G$ contains a strong parity factor $F$ if for every subset $X\subseteq V(G)$ with $|X|$ even, $G$ has a spanning subgraph $F$ satisfying $δ(F)\geq1$, $d_F(u)\equiv1$ (mod 2) for any $u\in X$, and $d_F(v)\equiv0$ (mod 2) for any $v\in V(G)\setminus X$. In this paper, we give a spectral radius condition to guarantee that a connected graph contains a strong parity factor.

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arXiv2024

Some sufficient conditions for graphs to have component factors

Let $G$ denote a graph and $k\geq2$ be an integer. A $\{K_{1,1},K_{1,2},\ldots,K_{1,k},\mathcal{T}(2k+1)\}$-factor of $G$ is a spanning subgraph, whose every connected component is isomorphic to an element of $\{K_{1,1},K_{1,2},\ldots,K_{1,k},\mathcal{T}(2k+1)\}$, where $\mathcal{T}(2k+1)$ is one special family of tree. In this paper, we put forward some sufficient conditions for the existence of $\{K_{1,1},K_{1,2},\ldots,K_{1,k},\mathcal{T}(2k+1)\}$-factors in graphs. Furthermore, we construct some extremal graphs to show that the main results in this paper are best possible.

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