Open Journal of Discrete Applied Mathematics (ODAM)

In this paper, we introduce a new resolvability parameter named as the local edge partition dimension \((LEPD)\) of graphs. The local edge partition dimension \((LEPD)\) makes a specialty of partitioning the vertex set of a graph into awesome instructions based totally on localized resolving properties. Our findings offer a fresh angle on graph resolvability, offering capability insights for optimizing network overall performance and structural analysis. Let \(G=(V, E)\) be a connected graph with vertex set \(V\) and edge set \(E\). A partition set \({R}_{p}=\{{R}_{p1},{R}_{p2},{R}_{p3}\dots,{R}_{pn}\}\) contain subsets of vertices of \(G\). If for every pair of adjacent edges \(p\) and \(q\) in \(G\), then \(d(p,{R}_{p})\neq d(q,{R}_{p})\) and if \(p\) and \(q\) are non-adjacent then not necessary \(d(p,{R}_{p})\neq d(q,{R}_{p})\) then \({R}_{p}\) is called a local edge resolving partition set and minimum cardinality of such set is called local edge partition dimension. We discussed local metric, local edge metric, metric, edge metric dimension, local partition, local edge partition, partition dimension, and edge partition dimension of the Petersen graph.

In this paper we construct families of bit sequences using combinatorial methods. Each sequence is derived by converting a collection of numbers encoding certain combinatorial numerics from objects exhibiting symmetry in various dimensions. Using the algorithms first described in [1] we show that the NIST testing suite described in publication 800-22 does not detect these symmetries hidden within these sequences.

This note addresses impracticalities or possible absurdities with regards to the definition corresponding of some graph parameters. To remedy the impracticalities the principle of transmitting the definition is put forward. The latter principle justifies a comprehensive review of many known graph parameters, the results related thereto, as well as the methodology of applications which draw a distinction between connected versus disconnected simple graphs. To illustrate the notion of transmitting the definition, various parameters are re-examined such as, connected domination number, graph diameter, girth, vertex-cut, edge-cut, chromatic number, irregularity index and quite extensively, the hub number of a graph. Ideas around undefined viz-a-viz permissibility viz-a-viz non-permissibility are also discussed.

A bijective mapping \(\varsigma\) assigns each vertex of a graph \(G\) a unique positive integer from 1 to \(|V(G)|\), with edge weights defined as the sum of the values at its endpoints. The mapping ensures that no two adjacent edges at a common vertex have the same weight, and each \(k\)-color class is connected to every other \(k-1\) color class. A graph \(G\) possesses \(b\)-color local edge antimagic coloring if it satisfies the aforementioned criteria and it corresponds to a maximum graph coloring. This paper extensively studies the bounds, non-existence, and results of b-color local edge antimagic coloring in fundamental graph structures.

Chemical graph theory, a branch of graph theory, uses molecular graphs for its representation. In QSAR/QSPR studies, topological indices are employed to evaluate the bioactivity of chemicals. Degree-based entropy, derived from Shannon’s entropy, is a functional statistic influenced by the graph and the probability distribution of its vertex set, with informational graphs forming the basis of entropy concepts. Planar octahedron networks have diverse applications in pharmacy hardware and system management. This article explores the Benzenoid Planar Octahedron Network (\(BPOH(n)\)), Benzenoid Dominating Planar Octahedron Network (BDPOH(n)), and Benzenoid Hex Planar Octahedron Network (\(BHPOH(n)\)). We compute degree-based entropies, including Randić entropy, atom bond connectivity (ABC), and geometric arithmetic (GA) entropy, for the Benzenoid planar octahedron network.

An edge irregular \(k\)-labeling of a graph \(G\) is a labeling of vertices of \(G\) with labels from the set \(\{1,2,3,\dots,k\}\) such that no two edges of \(G\) have same weight. The least value of \(k\) for which a graph \(G\) has an edge irregular \(k\)-labeling is called the edge irregularity strength of \(G\). Ahmad et. al. [1] showed the edge irregularity strength of some particular classes of Toeplitz graphs. In this paper we generalize those results and finds the exact values of the edge irregularity strength for some generalize classes of Toeplitz graphs.

The eccentric atom-bond sum-connectivity \(\left(ABSC_{e}\right)\) index of a graph \(G\) is defined as \(ABSC_{e}(G)=\sum\limits_{uv\in E(G)}\sqrt{\frac{e_{u}+e_{v}-2}{e_{u}+e_{v}}}\), where \(e_{u}\) and \(e_{v}\) represent the eccentricities of \(u\) and \(v\) respectively. This work presents precise upper and lower bounds for the \(ABSC_{e}\) index of graphs based on their order, size, diameter, and radius. Moreover, we find the maximum and minimum \(ABSC_{e}\) index of trees based on the specified matching number and the number of pendent vertices.

Let \(G = (V(G), E(G))\) be a graph with minimum degree at least \(1\). The inverse degree of \(G\), denoted \(Id(G)\), is defined as the sum of the reciprocals of degrees of all vertices in \(G\). In this note, we present inverse degree conditions for Hamiltonian and traceable graphs.

This note presents some upper bounds for the size of the upper deg-centric grapg \(G_{ud}\) of a simple connected graph G. Amongst others, a result for graphs for which a compliant graph \(G\) has \(G_{ud} \cong \overline G\) is presented. Finally, results for size minimality in respect upper deg-centrication and minimum size of such graph \(G\) are presented.

There are three different kinds of topological indices: spectrum-based, degree-based, and distance-based. We presented the \(K\)-swapped network for \(t\)-regular graphs in this study. We also computed various degree-based topological indices of the \(K\)-swapped network for \(t\)-regular graphs, eye, and \(n\)-dimensional twisted cube network. The metrics used to analyze the abstract structural characteristics of networks are called topological indices. We also calculate each of the aforementioned networks M-polynomials. A graph can be used to depict an interconnection network’s structure. The processing nodes in the network are represented by vertices, while the links connecting the processor nodes are represented by edges. We can quickly determine the diameter and degree between the nodes based on the graph’s tpology. A key component of graph theory are graph invariants, which identify the structural characteristics of networks and graphs. Furthermore doescribed by graph invariants are computer, social, and internet networks.