Related papers: Maximum spectral radius of graphs with given conne…

Let G be a graph of n vertices and m edges, and let G has no cycles of length 4. We give upper bounds on the adjacency spectral radius of G in terms of n and m.

Let $F_k$ be the (friendship) graph obtained from $k$ triangles by sharing a common vertex. The $F_k$-free graphs of order $n$ which attain the maximal spectral radius was firstly characterized by Cioab\u{a}, Feng, Tait and Zhang [Electron.…

A $k$-regular spanning subgraph of $G$ is called a $k$-factor. Fan, Lin and Lu [European J. Combin. 110 (2023) 103701] presented a tight sufficient condition in terms of the spectral radius for a connected 1-tough graph to contain a…

In this paper we show how the $\alpha$-spectral radius changes under the edge grafting operations on connected $k$-uniform hypergraphs. We characterize the extremal hypertree for $\alpha$-spectral radius among $k$-uniform non-caterpillar…

For $r\geq 3$, let $f_r\colon [0,\infty)\to [1,\infty)$ be the unique analytic function such that $f_r({k\choose r})={k-1\choose r-1}$ for any $k\geq r-1$. We prove that the spectral radius of an $r$-uniform hypergraph $H$ with $e$ edges is…

Let $\mathcal{D}_{n,\tau}$ be the set of all simple connected graphs of order $n$ and dissociation number $\tau.$ In this paper, we study the minimum size and the minimum spectral radius of graphs in $\mathcal{D}_{n,\tau}$ in connection…

The signless Laplacian spectral radius of a graph $G$, denoted by $q(G)$, is the largest eigenvalue of its signless Laplacian matrix. In this paper, we investigate extremal signless Laplacian spectral radius for graphs without short cycles…

A graph is non-trivial if it contains at least one nonloop edge. The essential connectivity of $G$, denoted by $\kappa'(G)$, is the minimum number of vertices of $G$ whose removal produces a disconnected graph with at least two components…

The $(k,r)$-fan is the graph consisting of $k$ copies of the complete graph $K_r$ which intersect in a single vertex, and is denoted by $F_{k,r}$. Erd\H{o}s, F\"uredi, Gould and Gunderson [J. Combin. Theory Ser. B 64 (1995) 89--100]…

For a graph $G$, its spectral radius is the largest eigenvalue of its adjacency matrix. A fan $H_{\ell}$ is a graph obtained by connecting a single vertex to all vertices of a path of order $\ell\geq4$. Let ${\rm SPEX(n,H_{\ell})}$ be the…

Let $G$ be a graph of order $n$ and spectral radius be the largest eigenvalue of its adjacency matrix, denoted by $\mu(G)$. In this paper, we determine the unique graph with maximum spectral radius among all graphs of order $n$ without…

Given a $k$-uniform hypergraph $G$ with vertex set $[n]$ and edge set $E(G)$, the ABC tensor $\mathcal{ABC}(G)$ of $G$ is the $k$-order $n$-dimensional tensor with \[ \mathcal{ABC}(G)_{i_1, \dots, i_k}= \begin{cases}…

We give a bound on the spectral radius of subgraphs of regular graphs with given order and diameter. We give a lower bound on the smallest eigenvalue of a nonbipartite regular graph of given order and diameter.

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 $k$-factor-critical if $G-S$ has a perfect matching for any $k$-subset $S$ of the vertex set of $G$. In this paper, we investigate the factor-criticality of graphs with fixed minimum degree and provide sufficient conditions…

The bandwidth theorem [Mathematische Annalen, 343(1):175--205, 2009] states that any $n$-vertex graph $G$ with minimum degree $(\frac{k-1}{k}+o(1))n$ contains all $n$-vertex $k$-colourable graphs $H$ with bounded maximum degree and…

Let $G$ be a connected graph. If $G$ contains a matching of size $k$, and every matching of size $k$ is contained in a perfect matching of $G$, then $G$ is said to be \emph{$k$-extendable}. A $k$-regular spanning subgraph of $G$ is called a…

Let $G=(V(G),E(G))$ be a simple graph, where $V(G)$ and $E(G)$ are the vertex set and the edge set of $G$, respectively. The number of components of $G$ is denoted by $c(G)$. Let $t$ be a positive real number, and a connected graph $G$ is…

The extremal graphs $\mathrm{EX}(n,\mathcal F)$ and spectral extremal graphs $\mathrm{SPEX}(n,\mathcal F)$ are the sets of graphs on $n$ vertices with maximum number of edges and maximum spectral radius, respectively, with no subgraph in…

In 1959, Erd\H{o}s and Gallai established two classic theorems, which determine the maximum number of edges in an $n$-vertex graph with no cycles of length at least $k$, and in an $n$-vertex graph with no paths on $k$ vertices,…