Related papers: Connected Hypergraphs with Small Spectral Radius

For a connected graph, the distance spectral radius is the largest eigenvalue of its distance matrix, and the distance energy is defined as the sum of the absolute values of the eigenvalues of its distance matrix. We establish lower and…

In this paper, we study the maximum spectral radius of outerplanar $3$-uniform hypergraphs. Given a hypergraph $\mathcal{H}$, the shadow of $\mathcal{H}$ is a graph $G$ with $V(G)= V(\mathcal{H})$ and $E(G) = \{uv: uv \in h \textrm{ for…

Let $\mathcal{A}(H)$ and $\mathcal{Q}(H)$ be the adjacency tensor and signless Laplacian tensor of an $r$-uniform hypergraph $H$. Denote by $\rho(H)$ and $\rho(\mathcal{Q}(H))$ the spectral radii of $\mathcal{A}(H)$ and $\mathcal{Q}(H)$,…

We give sharp upper bounds for the ordinary spectral radius and signless Laplacian spectral radius of a uniform hypergraph in terms of the average $2$-degrees or degrees of vertices, respectively, and we also give a lower bound for the…

The generalized distance spectral radius of a connected graph $G$ is the spectral radius of the generalized distance matrix of $G$, defined by $$D_\alpha(G)=\alpha Tr(G)+(1-\alpha)D(G), \;\;0\le\alpha \le 1,$$ where $D(G)$ and $Tr(G)$…

In this paper, we give the relationship between spectral radius and local structures of graphs and hypergraphs. Our work shows that certain local subgraphs (subhypergraphs) must occur when the spectral radius ratio is large. We also give…

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}…

For a connected graph $G$, let $A(G)$ be the adjacency matrix of $G$ and $D(G)$ be the diagonal matrix of the degrees of the vertices in $G$. The $A_{\alpha}$-matrix of $G$ is defined as \begin{align*} A_\alpha (G) = \alpha D(G) +…

Hypergraphs are an invaluable tool to understand many hidden patterns in large data sets. Among many ways to represent hypergraph, one useful representation is that of weighted clique expansion. In this paper, we consider this…

The spectral radius $\rho(G)$ of a graph $G$ is the largest eigenvalue of its adjacency matrix. Woo and Neumaier discovered that a connected graph $G$ with $\rho(G)\leq 3/2{\sqrt{2}}$ is either a dagger, an open quipu, or a closed quipu.…

Let $\mathcal{U}(n,k)$ and $\Gamma(n,k)$ be the set of the $k$-uniform linear and nonlinear unicyclic hypergraphs having perfect matchings with $n$ vertices respectively, where $n\geq k(k-1)$ and $k\geq 3$. By using some techniques of…

Identifying graphs with extremal properties is an extensively studied topic in spectral graph theory. In this paper, we study the log-concavity of a type of iteration sequence related to the $\alpha$-normal weighted incidence matrices which…

The spectral radius and rank of a graph are defined to be the spectral radius and rank of its adjacency matrix, respectively. It is an important problem in spectral extremal graph theory to determine the extremal graph that has the maximum…

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…

In this paper, we present several upper bounds for the adjacency and signless Laplacian spectral radii of uniform hypergraphs in terms of degree sequences.

Let $G$ be a graph with adjacency matrix $A(G)$ and degree diagonal matrix $D (G)$. In 2017, Nikiforov defined the matrix $A_\alpha(G) = \alpha D(G) + (1-\alpha)A(G)$ for any real $\alpha\in[0,1]$. The largest eigenvalue of $A_\alpha(G)$ is…

Let $G$ be a graph with adjacency matrix $A(G)$ and degree diagonal matrix $D (G)$. In 2017, Nikiforov [Appl. Anal. Discrete Math., 11 (2017) 81--107] defined the matrix $A_\alpha(G) = \alpha D(G) + (1-\alpha)A(G)$ for any real…

Let D be a strongly connected digraph and A(D) be the adjacency matrix of D. Let diag(D) be the diagonal matrix with outdegrees of the vertices of D and Q(D) = diag(D) + A(D) be the signless Laplacian matrix of D. The spectral radius of…

Let $G$ be a simple connected graph of size $m$. Let $A$ be the adjacency matrix of $G$ and let $\rho(G)$ be the spectral radius of $G$. A graph is said to be $H$-free if it does not contain a subgraph isomorphic to $H$. Let $H(\ell,3)$ be…

The best degree-based upper bound for the spectral radius is due to Liu and Weng. This paper begins by demonstrating that a (forgotten) upper bound for the spectral radius dating from 1983 is equivalent to their much more recent bound. This…