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A graph is $2$-planar if it has local crossing number two, that is, it can be drawn in the plane such that every edge has at most two crossings. A graph is maximal $2$-planar if no edge can be added such that the resulting graph remains…
The diameter of a graph is one if its most important parameters, being used in many real-word applications. In particular, the diameter dictates how fast information can spread throughout data and communication networks. Thus, it is a…
The eccentricity of a vertex $v$ in a graph $G$ is the maximum distance between $v$ and any other vertex of $G$. The diameter of a graph $G$ is the maximum eccentricity of a vertex in $G$. The eccentric connectivity index of a connected…
Given any two vertices u, v of a random geometric graph, denote by d_E(u,v) their Euclidean distance and by d_G(u,v) their graph distance. The problem of finding upper bounds on d_G(u,v) in terms of d_E(u,v) has received a lot of attention…
The topic of this paper is related to the well-known notion of unit distance graphs. Take a graph with its edges coloured red and blue such that for some $d$ it can be mapped into the plane with all vertices going to distinct points, the…
A bisection of a graph is a bipartition of its vertex set in which the number of vertices in the two parts differ by at most 1, and its size is the number of edges which go across the two parts. In this paper, motivated by several questions…
An adjacency-crossing graph is a graph that can be drawn such that every two edges that cross the same edge share a common endpoint. We show that the number of edges in an $n$-vertex adjacency-crossing graph is at most $5n-10$. If we…
Let $G$ be a connected graph. The average distance of a vertex $v$ of $G$ is the arithmetic mean of the distances from $v$ to all other vertices of $G$. The proximity and remoteness of $G$ are defined as the minimum and maximum,…
In this paper, we present a new metric distance for comparing two large graphs to find similarities and differences between them based on one of the most important graph structural properties, which is Node Adjacency Information, for all…
In this paper extremal values of the difference between several graph invariants related to the metric dimension are studied: mixed metric dimension, edge metric dimension and strong metric dimension. These non-trivial extremal values are…
We consider rectangle graphs whose edges are defined by pairs of points in diagonally opposite corners of empty axis-aligned rectangles. The maximum number of edges of such a graph on $n$ points is shown to be 1/4 n^2 +n -2. This number…
The edge betweenness centrality of an edge is loosely defined as the fraction of shortest paths between all pairs of vertices passing through that edge. In this paper, we investigate graphs where the edge betweenness centrality of edges is…
In this article, we establish some bounds involving the largest two distance Pareto eigenvalues of a connected graph. Also we characterize all possible values for smallest six distance Pareto eigenvalues of a connected graph.
A graph is a mathematical object consisting of a set of vertices and a set of edges connecting vertices. Graphs can be drawn on paper in various ways, but until recently all published methods of drawing graphs have had undesirable…
Vertex connectivity and edge connectivity are fundamental concepts in graph theory that have been widely studied from both structural and algorithmic perspectives. The focus of this paper is on computing these two parameters for graphs…
We study maximal distances in the commuting graphs of matrix algebras defined over algebraically closed fields. In particular, we show that the maximal distance can be attained only between two nonderogatory matrices. We also describe…
Edge lengths of a graph are called flexible if there exist infinitely many non-congruent realizations of the graph in the plane satisfying these edge lengths. It has been shown recently that a graph has flexible edge lengths if and only if…
The average distance of a vertex $v$ of a connected graph $G$ is the arithmetic mean of the distances from $v$ to all other vertices of $G$. The proximity $\pi(G)$ and the remoteness $\rho(G)$ of $G$ are the minimum and the maximum of the…
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…
Computer or communication networks are so designed that they do not easily get disrupted under external attack and, moreover, these are easily reconstructible if they do get disrupted. These desirable properties of networks can be measured…