Related papers: On trees with extremal extended spectral radius

The eccentricity matrix $\varepsilon(G)$ of a graph $G$ is constructed from the distance matrix of $G$ by keeping only the largest distances for each row and each column. This matrix can be interpreted as the opposite of the adjacency…

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 $G=(V,E)$ be a connected graph, where $V=\{v_1, v_2, \cdots, v_n\}$. Let $d_i$ denote the degree of vertex $v_i$. The ABC matrix of $G$ is defined as $M(G)=(m_{ij})_{n \times n}$, where $m_{ij}=\sqrt{(d_i + d_j -2)/(d_i d_j)}$ if $v_i…

For a graph $G=(V,E)$ and $v_{i}\in V$, denote by $d_{v_{i}}$ (or $d_{i}$ for short) the degree of vertex $v_{i}$. The $p$-Sombor matrix $\textbf{S}_{\textbf{p}}(G)$ ($p\neq0$) of a graph $G$ is a square matrix, where the $(i,j)$-entry is…

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…

For a graph $G$, let $\lambda_2(G)$ denote the second largest eigenvalue of the adjacency matrix of $G$. We determine the extremal trees with maximum/minimum adjacency eigenvalue $\lambda_2$ in the class $\mathcal{T}(n,d)$ of $n$-vertex…

The eccentricity matrix of a connected graph $G$, denoted by $\mathcal{E}(G)$, is obtained from the distance matrix of $G$ by keeping the largest nonzero entries in each row and each column, and leaving zeros in the remaining ones. The…

For a connected graph, the distance spectral radius is the largest eigenvalue of its distance matrix. In this paper, of all trees with both given order and fixed diameter, the trees with the minimal distance spectral radius are completely…

The spectral radius $\rho(G)$ of a graph $G$ is the largest eigenvalue of its adjacency matrix $A(G)$. For a fixed integer $e\ge 1$, let $G^{min}_{n,n-e}$ be a graph with minimal spectral radius among all connected graphs on $n$ vertices…

The spanning tree packing number of a graph $G$, denoted by $\tau(G)$, is the maximum number of edge-disjoint spanning trees contained in $G$. The study of $\tau(G)$ is one of the classic problems in graph theory. Cioab\u{a} and Wong…

For $0\le \alpha\le 1$, Nikiforov proposed to study the spectral properties of the family of matrices $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G)$ of a graph $G$, where $D(G)$ is the degree diagonal matrix and $A(G)$ is the adjacency matrix.…

Let G be a simple connected graph of order n with degree sequence d_1, d_2, ..., d_n in non-increasing order. The spectral radius rho(G) of G is the largest eigenvalue of its adjacency matrix. For each positive integer L at most n, we give…

The $f$ adjacency matrix is a type of edge-weighted adjacency matrix, whose weight of an edge $ij$ is $f(d_i,d_j)$, where $f$ is a real symmetric function and $d_i,d_j$ are the degrees of vertex $i$ and vertex $j$. The $f$-spectral radius…

Let $G$ be a connected graph and let $k$ be a positive integer. Let $T$ be a spanning tree of $G$. The leaf degree of a vertex $v\in V(T)$ is defined as the number of leaves adjacent to $v$ in $T$. The leaf degree of $T$ is the maximum leaf…

The eccentricity matrix $\varepsilon(G)$ of a graph $G$ is obtained from the distance matrix by retaining the eccentricities (the largest distance) in each row and each column. In this paper, we give a characterization of the star graph,…

Let $m$ be a positive integer. Brualdi and Hoffman proposed the problem to determine the (connected) graphs with maximum spectral radius in a given graph class and they posed a conjecture for the class of graphs with given size $m$. After…

Let $G$ be a graph with $n$ vertices and $m$ edges. The spectral radius $\rho(G)$ of $G$ is the largest eigenvalue of the adjacency matrix of $G$. As is well known, $\rho(G)\geq\frac{2m}{n}$ with equality if and only if $G$ is regular. To…

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 $\mathscr{G}_{n,\beta}$ be the set of graphs of order $n$ with given matching number $\beta$. Let $D(G)$ be the diagonal matrix of the degrees of the graph $G$ and $A(G)$ be the adjacency matrix of the graph $G$. The largest eigenvalue…

The eccentricity matrix of a connected graph $G$ is obtained from the distance matrix of $G$ by retaining the largest distances in each row and each column, and setting the remaining entries as $0$. In this article, a conjecture about the…