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Graph theory provides a primary tool for analyzing and designing computer communication networks. In the past few decades, Graph theory has been used to study various types of networks, including the Internet, wide Area Networks, Local Area…
The problem of connectivity assessment in an asymmetric network represented by a weighted directed graph is investigated in this article. A power iteration algorithm in a centralized implementation is developed first to compute the…
Matrix functions play an important role in applied mathematics. In network analysis, in particular, the exponential of the adjacency matrix associated with a network provides valuable information about connectivity, as well as about the…
We define the following parameter of connected graphs. For a given graph $G$ we place one agent in each vertex of $G$. Every pair of agents sharing a common edge is declared to be acquainted. In each round we choose some matching of $G$…
Graph theory is a promising approach in handling the problem of estimating the connectivity probability of vehicular ad-hoc networks (VANETs). With a communication network represented as graph, graph connectivity indicators become valid for…
A matrix $S=(s_{ij})\in{\mathbb R}^{n\times n}$ is said to determine a \emph{transitional measure} for a digraph $G$ on $n$ vertices if for all $i,j,k\in\{1,\...,n\},$ the \emph{transition inequality} $s_{ij} s_{jk}\le s_{ik} s_{jj}$ holds…
Typically, graph structures are represented by one of three different matrices: the adjacency matrix, the unnormalised and the normalised graph Laplacian matrices. The spectral (eigenvalue) properties of these different matrices are…
Finding shortest paths in a graph is relevant for numerous problems in computer vision and graphics, including image segmentation, shape matching, or the computation of geodesic distances on discrete surfaces. Traditionally, the concept of…
There are several centrality measures that have been introduced and studied for real world networks. They account for the different vertex characteristics that permit them to be ranked in order of importance in the network. Betweenness…
This article examines the application of a popular measure of sparsity, Gini Index, on network graphs. A wide variety of network graphs happen to be sparse. But the index with which sparsity is commonly measured in network graphs is edge…
Graph theory is increasingly commonly utilised in genetics, proteomics and neuroimaging. In such fields, the data of interest generally constitute weighted graphs. Analysis of such weighted graphs often require the integration of…
The energy of a graph $G$ is the sum of the absolute values of the eigenvalues of the adjacency matrix of $G$. Some variants of energy can also be found in the literature which are defined on the concepts of Laplacian matrix, Distance…
A number of network structural characteristics have recently been the subject of particularly intense research, including degree distributions, community structure, and various measures of vertex centrality, to mention only a few. Vertices…
Shortest paths are representative of discrete geodesic distances in graphs, and many descriptors of networks depend on their counting. In multiplex networks, this counting is radically important to quantify the switch between layers and it…
The use of complex networks as a modern approach to understanding the world and its dynamics is well-established in literature. The adjacency matrix, which provides a one-to-one representation of a complex network, can also yield several…
A fundamental problem in the study of complex networks is to provide quantitative measures of correlation and information flow between different parts of a system. To this end, several notions of communicability have been introduced and…
This article focuses on the problem of studying shared- and individual-specific structure in replicated networks or graph-valued data. In particular, the observed data consist of $n$ graphs, $G_i, i=1,\ldots,n$, with each graph consisting…
In recent years, graph neural networks (GNNs) have gained increasing popularity and have shown very promising results for data that are represented by graphs. The majority of GNN architectures are designed based on developing new…
For a graph $G$, the generalized adjacency matrix $A_\alpha(G)$ is the convex combination of the diagonal matrix $D(G)$ and the adjacency matrix $A(G)$ and is defined as $A_\alpha(G)=\alpha D(G)+(1-\alpha) A(G)$ for $0\leq \alpha \leq 1$.…
Traditional graph centrality measures effectively quantify node importance but fail to capture the structural uniqueness of multi-scale connectivity patterns -- critical for understanding network resilience and function. This paper…