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We study fundamental graph parameters such as the Diameter and Radius in directed graphs, when distances are measured using a somewhat unorthodox but natural measure: the distance between $u$ and $v$ is the minimum of the shortest path…
Given a connected graph $G=(V,E)$ and a vertex set $S\subset V$, the {\em Steiner distance} $d(S)$ of $S$ is the size of a minimum spanning tree of $S$ in $G$. For a connected graph $G$ of order $n$ and an integer $k$ with $2\leq k \leq n$,…
Let $G = (V, E)$ be a graph with non-empty set of vertices $V$ and set of edges $E$. The \emph{eccentric connectivity index} of the graph $G$ is defined as $$\displaystyle{\xi^C(G) = \sum_{u \in V} d_u \;ecc(u)}$$ where $d_u$ is the degree…
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…
By definition, a rigid graph in $\mathbb{R}^d$ (or on a sphere) has a finite number of embeddings up to rigid motions for a given set of edge length constraints. These embeddings are related to the real solutions of an algebraic system.…
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…
An outer-1-planar graph is a graph admitting a drawing in the plane so that all vertices appear in the outer region of the drawing and every edge crosses at most one other edge. This paper establishes the local structure of outer-1-planar…
The reciprocal degree resistance distance index of a connected graph $G$ is defined as $RDR(G)=\sum\limits_{\{u,v\}\subseteq V(G)}\frac {d_G(u)+d_G(v)}{r_G(u,v)}$, where $r_G(u,v)$ is the resistance distance between vertices $u$ and $v$ in…
A graph is outerplanar if it has a planar drawing for which all vertices belong to the outer face of the drawing. Let $H$ be a graph. The outerplanar Tur\'an number of $H$, denoted by $ex_\mathcal{OP}(n,H)$, is the maximum number of edges…
The Steiner distance of a graph, introduced by Chartrand, Oellermann, Tian and Zou in 1989, is a natural generalization of the concept of classical graph distance. For a connected graph $G$ of order at least $2$ and $S\subseteq V(G)$, the…
We consider drawings of graphs in the plane in which edges are represented by polygonal paths with at most one bend and the number of different slopes used by all segments of these paths is small. We prove that…
Recently, the problem of establishing bounds on the edge density of 1-planar graphs, including their subclass IC-planar graphs, has received considerable attention. In 2018, Angelini et al. showed that any n-vertex bipartite IC-planar graph…
A well known upper bound for the spectral radius of a graph, due to Hong, is that $\mu_1^2 \le 2m - n + 1$. It is conjectured that for connected graphs $n - 1 \le s^+ \le 2m - n + 1$, where $s^+$ denotes the sum of the squares of the…
We consider a random graph G(n,p) whose vertex set V has been randomly embedded in the unit square and whose edges are given weight equal to the geometric distance between their end vertices. Then each pair {u,v} of vertices have a distance…
This paper generalizes and unifies the existing spectral bounds on the $k$-independence number of a graph, which is the maximum size of a set of vertices at pairwise distance greater than $k$. The previous bounds known in the literature…
A 1-plane graph is a graph together with a drawing in the plane in such a way that each edge is crossed at most once. A 1-plane graph is maximal if no edge can be added without violating either 1-planarity or simplicity. Let $m(n)$ denote…
An orientation of $G$ is a digraph obtained from $G$ by replacing each edge by exactly one of two possible arcs with the same endpoints. We call an orientation \emph{proper} if neighbouring vertices have different in-degrees. The proper…
The eccentricity matrix $\epsilon(G)$, of a connected graph $G$ is obtained by retaining the maximum distance from each row and column of the distance matrix of $G$ and the other entries are assigned with 0. In this paper, we discuss the…
In a graph $G$, a vertex dominates itself and its neighbors. A subset $S\subseteq V(G)$ is said to be a double dominating set of $G$ if $S$ dominates every vertex of $G$ at least twice. The double domination number $\gamma_{\times 2}(G)$ is…
This paper discusses a distance guarding concept on triangulation graphs, which can be associated with distance domination and distance vertex cover. We show how these subjects are interconnected and provide tight bounds for any n-vertex…