On the spectral radius of clique trees with a given zero forcing number

Let $G(n,k)$ be the class of clique trees on $n$ vertices and zero forcing number $k$, where $\left \lfloor \frac{n}{2} \right \rfloor + 1 \le k \le n-1$ and each block is a clique of size at least $3$. In this article, we proved the existence and uniqueness of a clique tree in $G(n,k)$ that attains maximal spectral radius among all graphs in $G(n,k)$. We also provide an upper bound for the spectral radius of the extremal graph.Comment: 11 pages, 2 figures. arXiv admin note: text overlap with arXiv:2301.1279

On the spectral radius of block graphs with a given dissociation number

In this manuscript, we consider the spectral radius of graphs in the class of block graphs $\mathbf{Bl}(\textbf{k}, \varphi)$ with a fixed number of vertices $\textbf{k}$ and a given dissociation number $\varphi$. We first prove a few elementary results on the adjacency matrix of a graph, the spectral radius and a corresponding Perron vector. Using these results, we show a graph with maximal spectral radius in $\mathbf{Bl}(\textbf{k}, \varphi)$ satisfies a few necessary properties. These properties guarantee the existence and uniqueness of the block graph $\mathbb{B}_{\textbf{k},\varphi}$ that maximize the spectral radius in $\mathbf{Bl}(\textbf{k}, \varphi)$. Finally, we obtain bounds on the spectral radius of $\mathbb{B}_{\textbf{k},\varphi}$

Inverse of the Squared Distance Matrix of a Complete Multipartite Graph

Let $G$ be a connected graph on $n$ vertices and $d_{ij}$ be the length of the shortest path between vertices $i$ and $j$ in $G$. We set $d_{ii}=0$ for every vertex $i$ in $G$. The squared distance matrix $\Delta(G)$ of $G$ is the $n\times n$ matrix with $(i,j)^{th}$ entry equal to $0$ if $i = j$ and equal to $d_{ij}^2$ if $i \neq j$. For a given complete $t$-partite graph $K_{n_1,n_2,\cdots,n_t}$ on $n=\sum_{i=1}^t n_i$ vertices, under some condition we find the inverse $\Delta(K_{n_1,n_2,\cdots,n_t})^{-1}$ as a rank-one perturbation of a symmetric Laplacian-like matrix $\mathcal{L}$ with $\textup{rank} (\mathcal{L})=n-1$. We also investigate the inertia of $\mathcal{L}$.Comment: arXiv admin note: substantial text overlap with arXiv:2012.0434

On Squared Distance Matrix of Complete Multipartite Graphs

Let $G = K_{n_1,n_2,\cdots,n_t}$ be a complete $t$-partite graph on $n=\sum_{i=1}^t n_i$ vertices. The distance between vertices $i$ and $j$ in $G$, denoted by $d_{ij}$ is defined to be the length of the shortest path between $i$ and $j$. The squared distance matrix $\Delta(G)$ of $G$ is the $n\times n$ matrix with $(i,j)^{th}$ entry equal to $0$ if $i = j$ and equal to $d_{ij}^2$ if $i \neq j$. We define the squared distance energy $E_{\Delta}(G)$ of $G$ to be the sum of the absolute values of its eigenvalues. We determine the inertia of $\Delta(G)$ and compute the squared distance energy $E_{\Delta}(G)$. More precisely, we prove that if $n_i \geq 2$ for $1\leq i \leq t$, then $E_{\Delta}(G)=8(n-t)$ and if $h= |\{i : n_i=1\}|\geq 1$, then $$8(n-t)+2(h-1) \leq E_{\Delta}(G) < 8(n-t)+2h.$$ Furthermore, we show that for a fixed value of $n$ and $t$, both the spectral radius of the squared distance matrix and the squared distance energy of complete $t$-partite graphs on $n$ vertices are maximal for complete split graph $S_{n,t}$ and minimal for Tur{\'a}n graph $T_{n,t}$