A k-matching of a graph G is a function f:E(G)→{0,1,2,…,k} with ∑_e∈ E_G(v)f(e)≤ 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 ∑_e∈ 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)→ [0,1] satisfying ∑_e∈ E_G(v)f(e)=1 for any v∈ 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.

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 $d_F(v)\in\{0,2,\ldots,k+1\} \ \mbox{for all} \ v\in V(G)-W$.

Let G be a graph. We denote by e(G) and rho(G) the size and the spectral radius of G. A spanning subgraph F of G is called an even factor of G if dF(v) E {2, 4, 6, ...} for every v E 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 S contains an even factor with respect to its size and spectral radius. (c) 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Let G be a connected graph and T a spanning tree of G. Let rho(G) denote the adjacency The total k-excess te(T, k) is defined by te(T, k) = & sum; spectral radius of G. The k-excess of a vertex v in T is defined as max{0, dT(v)-k}. v is an element of V(T) max{0, dT(v)-k}. A tree T is said to be a k-tree if dT(v) <= k for any v is an element of 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 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 rho(G) in a connected graph G to ensure that G contains a spanning tree T with te(T, k) <= b, where k and b are two nonnegative integers with k >= max{5, b + 3} and (b, k)=/ (2, 5). (c) 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Let G be a connected graph with n vertices, where n is a positive integer. The size of G is denoted by I(G). The isolated toughness of G, denoted by I(G), is defined by I(G) = min {vertical bar S vertical bar/i(G - S ) : S subset of V(G) and i(G-S) >= 2 or I(G) = +infinity if G is complete. A graph G is called isolated r-tough if I(G) >= r. The distance signless Laplacian matrix Q(G) of G is defined by Q(G) = Tr(G) + D(G), where D(G) denotes the distance matrix of G and Tr(G) is the diagonal matrix of the vertex transmissions in G. The largest eigenvalue of Q(G), denoted by eta(G), is called the distance signless Laplacian spectral radius of G. A P->= k-factor means a path factor with every component containing at least k vertices, where k is an integer with k >= 2. In this paper, we aim to establish two tight sufficient conditions based on e(G) and eta(G) to guarantee that a graph G contains a P->= 2-factor. Let G be a connected isolated t 2/t+1-tough graph of order n, where t >= 1 is an integer. Then the following two results hold. (i) If n >= 6t + 2 and e(G) >= e(K-t V (Kn-3t-1 U (2t + 1)K-1)), then G contains a P->= 2-factor unless G = K-t V (Kn-3t-1 U (2t + 1)K-1). (ii) If n >= 9t + 2 and eta(G) <= eta(K-t V (Kn-3t-1 U (2t + 1)(K)1)), then G contains a P->= 2-factor unless G = K-t V (Kn-3t-1 U (2t + 1)K-1).

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)→ [0,1] such that ∑_e∈ E_G(v)h(e)=1 for every v∈ 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≥8k+6 and μ(G)(K_k∨(kK_1∪ K_3∪ K_n-2k-3)), then G contains a perfect matching unless G=K_k∨(kK_1∪ K_3∪ K_n-2k-3).

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)∈{1,3,5,…,b-1,b} for every v∈ V(G), where b≥2 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.

Let G be a connected graph with n vertices. The isolated toughness of G, denoted by I(G), is defined by I(G)=min{|S|/i(G-S):S⊆ V(G) i(G-S)≥2} if G is not complete, or I(G)=+∞ if G is complete. A graph G is called isolated r-tough if I(G)≥ r. A spanning subgraph H of G is called a {K_1,j:m≤ j≤2m}-factor of G if every component of H is isomorphic to an element of {K_1,j:m≤ j≤2m}. 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≥2. In this paper, we first establish a lower bounds on the adjacency spectral radius of a connected isolated mb-1/b-tough graph G to guarantees that G contains a {K_1,j:m≤ j≤2m}-factor. Second, we establish a lower bounds on the signless Laplacian spectral radius of a connected isolated mb-1/b-tough graph G to ensures that G contains a {K_1,j:m≤ j≤2m}-factor. Finally, we create an upper bounds on the distance spectral radius of a connected isolated mb-1/b-tough graph G with a {K_1,j:m≤ j≤2m}-factor. Furthermore, we construct some extremal graphs to claim that all the bounds obtained in this paper are sharp.

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) >= k for every v is an element of A was independently obtained by Frank and Gyarfas (A. Frank, E. Gyarfas, 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) >= k for every v is an element of A.

A spanning subgraph F of a graph G is called an odd [1,b]-factor if b equivalent to 1 (mod 2) and dF (v) is an element of {1, 3, ... , b} for every v is an element of V (G). A graph G of order n >= k + 2 is k-critical with respect to an odd [1, b]-factor if for any X subset of V (G) with |X| = k, G - X has an odd [1, b]factor. In this paper, we prove sharp lower bounds for both the size and spectral radius in an n-vertex (k + 1)-connected graph G with given minimum degree to guarantee that G is k-critical with respect to an odd [1, b]-factor.

The isolated toughness of a graph G, denoted by I(G), is defined by or I(G) = infinity if G is complete. A graph G is said to be isolated r-tough if I(G) >= r. A pathfactor of G is a spanning subgraph of G whose components are paths. Let P >= k = {Pi : i >= k >= 2}. A P >= k-factor means a path-factor in which every component is a path with at least k vertices. Liu, Lai and Das first introduced the matrix Aa(G) = aD(G) +A(G) of G [Spectral results on Hamiltonian problem, Discrete Math. 342 (2019) 1718-1730], where a >= 0 is an integer, and D(G) and A(G) respectively denote the diagonal degree matrix and the adjacency matrix of G. The largest eigenvalue of Aa(G), denoted by rho a(G), is called the Aa-spectral radius of G. The largest eigenvalue of the distance matrix D(G), denoted by & micro;(G), is called the distance spectral radius of G. In this paper, we aim to provide two sufficient conditions with respect to rho a(G) and & micro;(G) to guarantee the existence of P >= 2-factors in graphs. Let G be a connected isolated t 2t+1 -tough graph with n vertices, where t >= 1 is an integer. Then the following two results hold. (i) If n >= max{6t + 10, 2t2 + 5t + 4} and rho a(G) >= rho a(Kt boolean OR (Kn-3t-1 boolean OR (2t + 1)K1)) for a is an element of {0, 1}, then G has a P >= 2-factor unless G= Kt boolean OR (Kn-3t-1 boolean OR (2t + 1)K1). (ii) If n >= 9t + 2 and & micro;(G) <= & micro;(Kt boolean OR (Kn-3t-1 boolean OR (2t + 1)K1)), then G has a P >= 2-factor unless G = Kt boolean OR (Kn-3t-1 boolean OR (2t + 1)K1). (c) 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Let $\alpha\in[0,1)$, and let $G$ be a connected graph of order $n$ with $n\geq f(\alpha)$, where $f(\alpha)=14$ for $\alpha\in[0,\frac{1}{2}]$, $f(\alpha)=17$ for $\alpha\in(\frac{1}{2},\frac{2}{3}]$, $f(\alpha)=20$ for $\alpha\in(\frac{2}{3},\frac{3}{4}]$ and $f(\alpha)=\frac{5}{1-\alpha}+1$ for $\alpha\in(\frac{3}{4},1)$. A path factor is a spanning subgraph $F$ of $G$ such that every component of $F$ is a path with at least two vertices. Let $k\geq2$ be an integer. A $P_{\geq k}$-factor means a path-factor with each component being a path of order at least $k$. A graph $G$ is called a $P_{\geq k}$-factor covered graph if $G$ has a $P_{\geq k}$-factor containing $e$ for any $e\in E(G)$. Let $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G)$, where $D(G)$ denotes the diagonal matrix of vertex degrees of $G$ and $A(G)$ denotes the adjacency matrix of $G$. The largest eigenvalue of $A_{\alpha}(G)$ is called the $A_{\alpha}$-spectral radius of $G$, which is denoted by $\rho_{\alpha}(G)$. In this paper, it is proved that $G$ is a $P_{\geq2}$-factor covered graph if $\rho_{\alpha}(G)>\eta(n)$, where $\eta(n)$ is the largest root of $x^{3}-((\alpha+1)n+\alpha-4)x^{2}+(\alpha n^{2}+(\alpha^{2}-2\alpha-1)n-2\alpha+1)x-\alpha^{2}n^{2}+(5\alpha^{2}-3\alpha+2)n-10\alpha^{2}+15\alpha-8=0$. Furthermore, we provide a graph to show that the bound on $A_{\alpha}$-spectral radius is optimal.

Let G be a graph with vertex set V(G) and edge set E(G). For α∈[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 (G) of G is defined by (G)=min{|N_G(X)|/|X|:∅≠ X⊆ V(G),N_G(X)≠ V(G)}. If (G)≥1, 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)≤|S| holds for every S⊆ 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≥ n(α) has a perfect matching unless G=K_1∨(K_n-5∪ K_3∪ K_1) if ρ_α(G)_α(K_1∨(K_n-5∪ K_3∪ K_1)), where n(α) is defined as follows: n(α)=max{18,2+8α/1-2α} if α∈[0,1/2), and n(α)=18 if α=1/2.

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{|S|/c(G-S):S⊆ V(G), c(G-S)≥2} for G≠ K_n and t(G)=+∞ 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{|S|/c(G-S):S⊂ V(G), c(G-S)≥ l} if 2≤ l(G), and t_l(G)=+∞ if l>α(G). If t_l(G)≥ 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 (1/b,l)-tough, respectively.

Let G be a graph and k >= 4 be an integer. A {K1,1, K1,2, ..., K1,k, T(2k + 1)}-factor of G is a spanning subgraph, every connected component of which is isomorphic to a member of {K1,1, K1,2, ..., K1,k, T(2k + 1)}, where T(2k + 1) is one special family of trees. In this paper, we put forward a sufficient spectral condition for a graph to contain a {K1,1, K1,2, ..., K1,k, T(2k+1)}-factor. Furthermore, we construct three extremal graphs to claim that the bounds on the spectral radius in our main result are sharp. (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

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].

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 $\rho(G)$ 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 $\rho(G)\geq\rho(K_1\vee(K_{n-4}\cup K_2\cup K_1))$, then $G$ has an $H$-factor for each $H:V(G)\longrightarrow\{\{1\},\{0,2\}\}$ with $H^{-1}(1)$ even, unless $G=K_1\vee(K_{n-4}\cup K_2\cup K_1)$.

Let m, t, r and ki (1 <= i <= m) be positive integers with ki >= (2r-1)t + 1. Let G be a graph, H be an mr-subgraph of G, and F = {F-1, F-2, ... , Fm} be a (g, f)-factorization of G. If for any partition {A(1), A(2), ... , A(m)} of E(H) with |A(i )|= r, G has a (g, f )-factorization F = {F-1, F-2, ... , Fm} with A(i )subset of E(F-i), 1 <= i <= m, then we say that G has (g, f)-factorizations randomly r-orthogonal to H. Let H-1, H-2, ... , H-t be t vertex-disjoint mr-subgraphs of a bipartite graph G with Delta(G) <= k(1) +k(2) + +k(m)-m+1. We demonstrate that a bipartite graph G with A(G) <= k(1) +k(2 )+ +k(m)-m+1 possesses a [0, ki](m)(1)-factorization randomly r-orthogonal to every H-i, 1 <= i <= t.

Let $k$ and $n$ be two nonnegative integers with $n\equiv0$ (mod 2), and let $G$ be a graph of order $n$ with a 1-factor. Then $G$ is said to be $k$-extendable for $0\leq k\leq\frac{n-2}{2}$ if every matching in $G$ of size $k$ can be extended to a 1-factor. In this paper, we first establish a lower bound on the signless Laplacian spectral radius of $G$ to ensure that $G$ is $k$-extendable. Then we create some extremal graphs to claim that all the bounds derived in this article are sharp.

The binding number of a graph G, written as (G), is defined by (G)=min{|N_G(X)|/|X|:∅≠ X⊆ V(G),N_G(X)≠ V(G)}. A graph G is called r-binding if (G)≥ r. An odd [1,b]-factor of a graph G is a spanning subgraph F with d_F(v)∈{1,3,…,b} for all v∈ V(G), where b≥1 is an odd integer. A spanning k-tree of a connected graph G is a spanning tree T with d_T(v)≤ k for every v∈ V(G). In this paper, we first show a tight sufficient condition with respect to the adjacency spectral radius for connected 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 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].