Related papers: Characterizing graphs with the second largest dist…

For a fixed positive integer $k$ and a graph $G$, let $\lambda_k(G)$ denote the $k$-th largest eigenvalue of the adjacency matrix of $G$. In 2017, Tait and Tobin proved that the maximum $\lambda_1(G)$ among all outerplanar graphs on $n$…

Let $G=(V(G),E(G))$ be a simple graph, and let $U\subseteq V(G)$. Two distinct vertices $x,y\in U$ are $U$-mutually visible if $G$ contains a shortest $x$-$y$ path that is internally disjoint from $U$. $U$ is called a mutual-visibility set…

In this paper, we consider the distance spectra of the derangement graphs. First we give a constructive proof that the connected derangement graphs are of diameter 2. Then we obtain their distance spectra. In particular, we determine all…

For a graph $G$ and $a,b\in V(G)$, the shortest path reconfiguration graph of $G$ with respect to $a$ and $b$ is denoted by $S(G,a,b)$. The vertex set of $S(G,a,b)$ is the set of all shortest paths between $a$ and $b$ in $G$. Two vertices…

The normalized distance Laplacian of a graph $G$ is defined as $\mathcal{D}^\mathcal{L}(G)=T(G)^{-1/2}(T(G)-\mathcal{D}(G))T(G)^{-1/2}$ where $\mathcal{D}(G)$ is the matrix with pairwise distances between vertices and $T(G)$ is the diagonal…

Let $G$ be a simple connected graph of order $n$ and $\partial(G)$ is the spectral radius of the distance matrix $D(G)$ of $G$. The transmission $D_i$ of vertex $i$ is the $i$-th row sum of $D(G)$. Denote by $D_{\max}(G)$ the maximum of…

The second largest eigenvalue of a graph is an important algebraic parameter which is related with the expansion, connectivity and randomness properties of a graph. Expanders are highly connected sparse graphs. In coding theory, Expander…

For a graph $G$, and two distinct vertices $u$ and $v$ of $G$, let $n_G(u,v)$ be the number of vertices of $G$ that are closer in $G$ to $u$ than to $v$. Miklavi\v{c} and \v{S}parl (arXiv:2011.01635v1) define the distance-unbalancedness of…

Let $\delta$ and $\Delta$ be the minimum and the maximum degree of the vertices of a simple connected graph $G$, respectively. The distinguishing index of a graph $G$, denoted by $D'(G)$, is the least number of labels in an edge labeling of…

For a connected graph $G$ of order at least $2$ and $S\subseteq V(G)$, the \emph{Steiner distance} $d_G(S)$ among the vertices of $S$ is the minimum size among all connected subgraphs whose vertex sets contain $S$. In this paper, we…

The eccentric connectivity index of a connected graph $G$ is the sum over all vertices $v$ of the product $d_{G}(v) e_{G}(v)$, where $d_{G}(v)$ is the degree of $v$ in $G$ and $e_{G}(v)$ is the maximum distance between $v$ and any other…

For a connected graph $G$ of order at least $2$ and $S\subseteq V(G)$, the \emph{Steiner distance} $d_G(S)$ among the vertices of $S$ is the minimum size among all connected subgraphs whose vertex sets contain $S$. Let $n$ and $k$ be two…

In this article, we study Pareto eigenvalues of distance matrix of connected graphs and show that the non zero entries of every distance Pareto eigenvector of a tree forms a strictly convex function on the forest generated by the vertices…

Given a regular (connected) graph $\Gamma=(X,E)$ with adjacency matrix $A$, $d+1$ distinct eigenvalues, and diameter $D$, we give a characterization of when its distance matrix $A_D$ is a polynomial in $A$, in terms of the adjacency…

A vertex $v \in V(G)$ is called $\lambda$-main if it belongs to a star set $X \subset V(G)$ of the eigenvalue $\lambda$ of a graph $G$ and this eigenvalue is main for the graph obtained from $G$ by deleting all the vertices in $X \setminus…

Let $G$ be a connected tree on $n$ vertices and let $L = D-A$ denote the Laplacian matrix on $G$. The second-smallest eigenvalue $\lambda_{2}(G) > 0$, also known as the algebraic connectivity, as well as the associated eigenvector $\phi_2$…

For a finite, simple, and undirected graph $G$ with $n$ vertices, $m$ edges, and largest eigenvalue $\lambda$, Nikiforov introduced the degree deviation of $G$ as $s=\sum_{u\in V(G)}\left|d_G(u)-\frac{2m}{n}\right|$. Contributing to a…

The distinguishing number $D(G)$ of a graph $G$ is the least integer $d$ such that $G$ has an vertex labeling with $d$ labels that is preserved only by a trivial automorphism. A list assignment to $G$ is an assignment $L = \{L(v)\}_{v\in V…

The min-diameter of a directed graph $G$ is a measure of the largest distance between nodes. It is equal to the maximum min-distance $d_{min}(u,v)$ across all pairs $u,v \in V(G)$, where $d_{min}(u,v) = \min(d(u,v), d(v,u))$. Our work…

Let $G$ be a simple connected graph. If every pendant path in $G$ is at least $P_s$, we denote that $G\in \mathbb{G}_s$. For $G \in \mathbb{G}_s$, let $Q_s(G)$ be the set of vertices in $G$ that are distance $s$ from the pendant vertex, and…