Partition-based construction and stability analysis of Euler graphs using vertex strength

Evaluating the Influence of Graph Density on the Efficiency of Shortest Path Algorithms Using Different Data Structures

This paper presents a comparative analysis of two variants of a classical algorithm for finding the shortest path in a connected graph. The first variant uses an adjacency matrix (AM) to verify the existence of an edge (arc) between two vertices, while the second variant performs the same verification using an adjacency list (AL). The objective of this study is to examine how graph density affects the performance of the two algorithmic modifications depending on the data structure used. A total of 95 graphs were analyzed, grouped into five sets from 100 to 500 in increments of 100. For each group, 19 graphs were generated with densities ranging from 5% to 95% in increments of 5%. The methodology includes analyzing the number of iterations, assignments, and comparisons executed by the algorithms for all graphs. The initial hypothesis assumed that the total number of operations would always be lower when using an AL instead of an adjacency matrix, regardless of graph density. The results demonstrate that this assumption is incorrect: for densities above 82%, the total number of operations is lower when using an adjacency matrix, whereas the AL is more efficient for densities below 82%, with its efficiency increasing as density decreases. These findings are particularly important for mobile technologies, as they support the design of more efficient pathfinding solutions that optimize performance and energy consumption in mobile applications.

A Spectral Approach to Join Based Operations on Graphs

This study explores the spectral characteristics and energy distributions associated with selected graph operations derived from the first Zagreb, second Zagreb, and sum-connectivity matrices to contribute to understanding how algebraic operations induce spectral energy shifts analogous to perturbations in physical or molecular graph systems.

Permanent fuzzy graphs with application in decision making

Steiner Geodesic Distance in Fuzzy Graphs with applications in Telecommunication Networks

Distance in fuzzy graphs is a fundamental concept crucial in analyzing connectivity patterns and network dynamics. This article aims to extend and generalize the notion of distance beyond pair of vertices by introducing the novel concept of Steiner geodesic distance (SGD) in fuzzy graphs, which serves as a broader extension of the geodesic distance. If a fuzzy graph has n vertices, this approach provides a framework for determining the distance among k vertices, where [Formula: see text], thereby laying the foundation for analyzing interactions among multiple vertices in the network. The Steiner geodesic tree is defined and the bounds of SGD are obtained. An algorithm for finding Steiner geodesic (SG) in [Formula: see text] running time is presented. The notions of SG k - eccentricity, SG k - center, SG k - selfcentered fuzzy graphs and SG k - eccentric set are established and a characterization of SG k - selfcentered fuzzy graphs is provided. The SGD concepts are analyzed on fuzzy graphs including complete fuzzy graph, complete bipartite fuzzy graph, fuzzy cycle and fuzzy trees. The idea of SGD k - matrix is also discussed and proposed an algorithm to distinguish between SG k - selfcentered fuzzy graphs. The article also presents applications of SGD concepts in optimizing fiber optic cable routing and identifying the optimal location for a central command hub in encrypted radio communication networks.

On the Extra Connectivity of the Power Graph of a Finite Cyclic Group

Reliability evaluation of an interconnection network is of great significance for construction and maintenance of the network. The extra connectivity and essentially edge-connectivity are two important parameters to evaluate network reliability. Let [Formula: see text] be a finite group. The power graph [Formula: see text] of [Formula: see text] is defined as an undirected graph whose vertex set is [Formula: see text] and two distinct vertices [Formula: see text] are adjacent if and only if one is a power of the other. In this paper, we determine the [Formula: see text]-extra connectivity and essentially edge-connectivity of the power graph of a cyclic finite group.

Pseudo A4-graphs and their properties in dihedral groups

Let G be a finite group. We introduce a new graph definition, called the pseudo A4-graph, as an extension of the classical A4-graph. The pseudo A4-graph of G or PA4-graph is a simple graph ΓG whose vertices are elements of G, and two distinct vertices a and b are joined by an edge if and only if ab−1 = ba−1. This paper presents the formulas for certain graph invariants such as the number of edges, diameter, total degree, chromatic number, clique number, independence number, domination number, matching number, and graph energy. The method involves constructing the A4-graphs of dihedral groups, analyzing their structure, and systematically computing graph invariants through algebraic reasoning and combinatorial techniques to establish fundamental properties and relationships. One of the results in this paper comply with the well-known fact that the energy of a graph is always an even integer.