Related papers: The $A_\alpha$ spectral radius with given independ…

Given a simple graph $G$, its $A_\alpha$ matrix is a convex combination with parameter $\alpha\in [0,1]$ of its adjacency matrix and its degree diagonal matrices. Here we compare two lower bounds presented in [J. D. G. Silva Jr., C. S.…

Let $G$ be a graph. The spectral radius $\rho(G)$ of $G$ is the largest eigenvalue of its adjacency matrix. For an integer $k\geq1$, a $k$-factor of $G$ is a $k$-regular spanning subgraph of $G$. Assume that $k$ and $n$ are integers…

In 2017, Nikiforov introduced the concept of the $A_{\alpha}$-matrix, as a linear convex combination of the adjacency matrix and the degree diagonal matrix of a graph. This matrix has attracted increasing attention in recent years, as it…

Let $A(G)$ be the adjacent matrix and $D(G)$ the diagonal matrix of the degrees of a graph $G$, respectively. For $0 \leq \alpha \leq 1$, the $A_{\alpha}$ matrix $A_{\alpha}(G) = \alpha D(G) +(1-\alpha)A(G)$ is given by Nikiforov. Clearly,…

For a graph $G$, the spectral radius $\rho(G)$ of $G$ is the largest eigenvalue of its adjacency matrix. In this paper, we give three lammas on $\rho(G)$ when $G$ contains a spanning complete bipartite graph. Using these lemmas and typical…

Let $G$ be a connected simple graph with $n$ vertices. The distance Laplacian matrix $D^{L}(G)$ is defined as $D^L(G)=Diag(Tr)-D(G)$, where $Diag(Tr)$ is the diagonal matrix of vertex transmissions and $D(G)$ is the distance matrix of $G$.…

For a connected graph $G$ and $\alpha\in [0,1)$, the distance $\alpha$-spectral radius of $G$ is the spectral radius of the matrix $D_{\alpha}(G)$ defined as $D_{\alpha}(G)=\alpha T(G)+(1-\alpha)D(G)$, where $T(G)$ is a diagonal matrix of…

Suppose that $G$ is a connected simple graph with the vertex set $V( G ) = \{ v_1,v_2,\cdots ,v_n \} $. Let $d( v_i,v_j ) $ be the distance between $v_i$ and $v_j$. Then the distance matrix of $G$ is $D( G ) =( d_{ij} )_{n\times n}$, where…

The arithmetic-geometric matrix $A_{ag}(G)$ of a graph $G$ is a square matrix, where the $(i,j)$-entry is equal to $\displaystyle \frac{d_{i}+d_{j}}{2\sqrt{d_{i}d_{j}}}$ if the vertices $v_{i}$ and $v_{j}$ are adjacent, and 0 otherwise. The…

Let $\mathbb{Q}_{k,n}$ be the set of the connected $k$-uniform weighted hypergraphs with $n$ vertices, where $k,n\geq 3$. For a hypergraph $G\in \mathbb{Q}_{k,n}$, let $\mathcal{A}(G)$, $\mathcal{L} (G)$ and $\mathcal{Q} (G)$ be its…

Let $Q(G)=D(G)+A(G)$ be the signless Laplacian matrix of a simple graph $G$, where $D(G)$ and $A(G)$ are the degree diagonal matrix and the adjacency matrix of $G$, respectively. The largest eigenvalue of $Q(G)$, denoted by $q(G)$, is…

Brualdi and Hoffman proposed a well-known problem of determining the graph with maximum adjacency spectral radius among all graphs with given size $m$. Early work by Friedland and Stanley addressed some specific cases. This problem was…

Let G be a graph with n vertices and mu(G) be the largest eigenvalue of the adjacency matrix of G. We study how large mu(G) can be when G does not contain cycles and paths of specified order. In particular, we determine the maximum spectral…

We give an upper bound on the maximal eigenvalue of the adjacency matrix of a connected graph in terms of its maximum degree, diameter and order. This bound is best possible up to a constant factor and improves prevoius results of…

The spectral radius of a graph is the largest eigenvalue of its adjacency matrix. A minimizer graph is such that minimizes the spectral radius among all connected graphs on $n$ vertices with diameter $d$. The minimizer graphs are known for…

It is well known that spectral Tur\'{a}n type problem is one of the most classical {problems} in graph theory. In this paper, we consider the spectral Tur\'{a}n type problem. Let $G$ be a graph and let $\mathcal{G}$ be a set of graphs, we…

For a graph $G$, the spectral radius of $G$ is the largest eigenvalue of its adjacency matrix. A connected factor of $G$ is a connected spanning subgraph of $G$. For example, a spanning tree of $G$ is a 1-connected factor of $G$. Let $G$ be…

The weighted adjacency matrix $A_{f}(G)$ of a simple graph $G=(V,E)$ is the $|V|\times|V|$ matrix whose $ij$-entry equals $f(d_{i},d_j)$, where $f(x,y)$ is a symmetric function such that $f(d_i,d_j)>0$ if $ij\in E$ and $f(d_i,d_j)=0$ if…

Let $G$ be a graph attaining the maximum spectral radius among all connected nonregular graphs of order $n$ with maximum degree $\Delta$. Let $\lambda_1(G)$ be the spectral radius of $G$. A nice conjecture due to Liu, Shen and Wang [On the…

For a $hypergraph$ $\mathcal{G}=(V, E)$ consisting of a nonempty vertex set $V=V(\mathcal{G})$ and an edge set $E=E(\mathcal{G})$, its $adjacency$ $matrix$ $\mathcal {A}_{\mathcal{G}}=[(\mathcal {A}_{\mathcal{G}})_{ij}]$ is defined as…