Related papers: Toughness and A{\alpha}-spectral radius in graphs

Given a connected graph $G$, the toughness $\tau_G$ is defined as the minimum value of the ratio $|S|/\omega_{G-S}$, where $S$ ranges over all vertex cut sets of $G$, and $\omega_{G-S}$ is the number of connected components in the subgraph…

Let $G$ be a simple, connected graph and let $A(G)$ be the adjacency matrix of $G$. If $D(G)$ is the diagonal matrix of the vertex degrees of $G$, then for every real $\alpha \in [0,1]$, the matrix $A_{\alpha}(G)$ is defined as…

A path in an edge colored graph is said to be a rainbow path if no two edges on the path have the same color. An edge colored graph is (strongly) rainbow connected if there exists a (geodesic) rainbow path between every pair of vertices.…

Let $G$ be a graph. We say that a hypergraph $H$ is a Berge-$G$ if there is a bijection $\phi: E(G)\to E(H)$ such that $e\subseteq \phi(e)$ for all $e\in E(G)$. For any $r$-uniform hypergraph $H$ and a real number $p\geq 1$, the…

For a graph $G=(V,E)$ and $v_{i}\in V$, denote by $d_{i}$ the degree of vertex $v_{i}$. Let $f(x, y)>0$ be a real symmetric function in $x$ and $y$. The weighted adjacency matrix $A_{f}(G)$ of a graph $G$ is a square matrix, where the…

Let $\mathcal{G}=\{G_1,\ldots, G_{\frac{kn}{2}}\}$ be a set of graphs on the same vertex set $V=\{1,\dots,n\}$ where $k\cdot n$ is even. We say $\mathcal{G}$ admits a rainbow $k$-factor if there exists a $k$-regular graph $F$ on the vertex…

A graph is called traceable if it contains a Hamilton path, i.e., a path passing through all its vertices. Let $G$ be a graph on $n$ vertices. $G$ is called claw-$o_{-1}$-heavy if every induced claw ($K_{1,3}$) of $G$ has a pair of…

A path in an edge colored graph is said to be a rainbow path if no two edges on the path have the same color. An edge colored graph is (strongly) rainbow connected if there exists a (geodesic) rainbow path between every pair of vertices.…

Let $\mathbb{G}_{n,\alpha}$ be the set of connected graphs with order $n$ and independence number $\alpha$. Given $k=n-\alpha$, the graph with minimum spectral radius among $\mathbb{G}_{n,\alpha}$ is called the minimizer graph.…

Let $\mu(G)$ denote the spectral radius of a graph $G$. We partly confirm a conjecture due to Nikiforov, which is a spectral radius analogue of the well-known Erd\H{o}s-S\'os Conjecture that any tree of order $t$ is contained in a graph of…

Let $G$ be a simple connected graph with vertex set $V(G)$ and edge set $E(G)$. A $k$-matching of a graph $G$ is a function $f:E(G)\rightarrow \{0,1,\ldots, k\}$ satisfying $\sum_{e \in E_G(v)} f(e) \leq k$ for every vertex $v \in V(G)$,…

Let $G$ be a graph on $n$ vertices and $m$ edges. For $\alpha \in [0,1]$, the $A_{\alpha}$-matrix of $G$ is defined as $A_{\alpha}(G) = \alpha D(G) + (1- \alpha) A(G)$, where $A(G)$ is the adjacency matrix and $D(G)$ is the degree diagonal…

Let $G=(V,E)$ be a connected graph, where $V=\{v_1, v_2, \cdots, v_n\}$ and $m=|E|$. $d_i$ will denote the degree of vertex $v_i$ of $G$, and $\Delta=\max_{1\leq i \leq n} d_i$. The ABC matrix of $G$ is defined as $M(G)=(m_{ij})_{n \times…

A graph $ G $ is called $ t $-tough if $ \left|S\right|\geq t\cdot w\left(G-S\right)$ for every cutset $ S $ of $G$. Chv\'atal conjectured that there exists a constant $ t_{0} $ such that every $ t_{0} $-tough graph has a hamiltonian cycle.…

Let $\mathcal{G}_{n, \beta^*}$ $(\mathcal{G}^*_{n,\beta^*})$ be the set of all (connected) graphs of order $n$ with fractional matching number $\beta^*$. In this paper, the graphs with maximal spectral radius in $\mathcal{G}_{n,\beta^*}$…

A graph is {\em $1$-planar} if it can be drawn in the plane such that every edge crosses at most one other edge. A connected graph $H$ is {\em strongly light} in a family of graphs $\mathfrak{G}$, if there exists a constant $\lambda$, such…

Let $\tau(G)$ denote the spanning tree packing number of a graph $G$. Recently, Zhang and Fan [J. Graph Theory 112 (2) (2026) 128--144] posed the problem of finding a tight spectral radius condition for an $m$-edge-connected graph $G$ to…

The thickness of a graph $G=(V,E)$ with $n$ vertices is the minimum number of planar subgraphs of $G$ whose union is $G$. A polyline drawing of $G$ in $\mathbb{R}^2$ is a drawing $\Gamma$ of $G$, where each vertex is mapped to a point and…

A graph $ G $ is minimally $ t $-tough if the toughness of $ G $ is $ t $ and deletion of any edge from $ G $ decreases its toughness. Katona et al. conjectured that the minimum degree of any minimally $ t $-tough graph is $ \lceil 2t\rceil…

In a graph $G$, a subset of vertices $S \subseteq V(G)$ is said to be cyclable if there is a cycle containing the vertices in some order. $G$ is said to be $k$-cyclable if any subset of $k \geq 2$ vertices is cyclable. If any $k$…