Saima Nazeer

The concept of graph energy, first introduced in 1978, has been a focal point of extensive research within the field of graph theory, leading to the publication of numerous articles. Graph energy, originally associated with the eigenvalues of the adjacency matrix of a graph, has since been extended to various other matrices. These include the maximum degree matrix, Randić matrix, sum-connectivity matrix, and the first and second Zagreb matrices, among others. In this paper, we focus on calculating the energy of several such matrices for the join graph of complete graphs, denoted as J m ( K n ) . Specifically, we compute the energies for the maximum degree matrix, Randić matrix, sum-connectivity matrix, first Zagreb matrix, second Zagreb matrix, reverse first Zagreb matrix, and reverse second Zagreb matrix for J m ( K n ) . Our results provide new insights into the structural properties of the join graph and contribute to the broader understanding of the mathematical characteristics of graph energy for different matrix representations. This work extends the scope of graph energy research by considering these alternative matrix forms, offering a deeper exploration into the algebraic and spectral properties of graph energy in the context of join graphs.

Graph theory and its wide applications in natural sciences and social sciences open a new era of research. Making the graph of computer networks and analyzing it with aid of graph theory are extensively studied and researched in the literature. An important discussion is based on distance between two nodes in a network which may include closeness of objects, centrality of objects, average path length between objects, and vertex eccentricity. For example, (1) disease transmission networks: closeness and centrality of objects are used to measure vulnerability to particular disease and its infectivity; (2) routing networks: eccentricity of objects is used to find vertices which form the periphery objects of the network. In this manuscript, we have discussed distance measurements including center, periphery, and average eccentricity for the Cartesian product of two cycles. The results are obtained using the definitions of eccentricity, radius, and diameter of a graph, and all possible cases (for different parity of length of cycles) have been proved.

The graph entropy was proposed by Körner in the year 1973 when he was studying the problem of coding in information theory. The foundation of graph entropy is in information theory, but it was demonstrated to be firmly identified with some established and often examined graph-theoretic ideas. For instance, it gives an equal definition to a graph to be flawless, and it can likewise be connected to acquire lower bounds in graph covering problems. The objective of this study is to solve the open problem suggested by Kwun et al. in 2018. In this paper, we study the weighted graph entropy by taking augmented Zagreb edge weight and give bounds of it for regular, connected, bipartite, chemical, unicyclic, etc., graphs. Moreover, we compute the weighted graph entropy of certain nanotubes and plot our results to see dependence of weighted entropy on involved parameters.

In the fields of mathematical chemistry and chemical graph theory, a topological index generally called a connectivity index is a kind of a molecular descriptor that is calculated in perspective of the molecular graph of a chemical compound. Topological indices are numerical parameters of a graph which depict its topology and are graph invariant up to graph isomorphism. Topological indices are used for example in the progression of quantitative structure-activity relationships (QSARs) in which the common activity or distinctive properties of atoms are connected with their molecular structure. There are in excess of 140 topological indices but none of them totally describe the molecular compound completely so there is dependably a space to characterize and register new topological indices. Benzenoid Systems are utilized basically as an intermediate to make different synthetic compounds. In this report we aim to compute redefined Zagreb indices for Zigzag, Rhombic, triangular, Hourglass and Jagged-rectangle Benzenoid systems.

Many variations of graph labeling has been defined in the literature. e.g., graceful, harmonious and radio labeling etc. In information technology and in data sciences, we need secrecy of data, different channel assignment and accuracy of transmission of the data. This make the use of graph terminologies indispensable for the computer programs. In this paper, we will discuss multi-distance radio labeling used for channel assignment problem over a wireless communication. A radio (multi-distance) labeling of a graph $G$ is a function $h$ from $V(G)$ to the set of non-negative integers such that $|h(u)-h(v)| \geq diam(G)+1- d_{G}(u,v)$ ,Where $diam(G)$ and $d_{G}(u,v)$ are diameter and distance between $u$ and $v$ in graph $G$ respectively. The span of a radio labeling $h$ is the maximum integer assigned by $h$ and radio number of a graph $G$ is the minimum span taken over all radio labeling of $G$ . In this article, we will find relations for radio number of a strong product $K_{3} \boxtimes P _{n},~n \geq 3$ .

Let G be a connected graph and d(μ,ω) be the distance between any two vertices of G. The diameter of G is denoted by diam(G) and is equal to {d(μ,ω); μ,ω ∈ G}. The radio labeling(RL) for the graph G is an injective function ϝ : V(G) → N ⋃ {0} such that for any pair of vertices μ and ω |ϝ(μ) – ϝ(ω)|≥diam(G) – d(μ,ω)+1. The span of radio labeling is the largest number in ϝ(V). The radio number of G, denoted by rn(G) is the minimum span over all radio labeling of G. In this paper, we determine radio number for the generalized Petersen graphs, P(n,2), n = 4k+2. Further the lower bound of radio number for P(n,2) when n = 4k is determined.