The energy of a simple connected graphGis equal to the sum of the absolute value of eigenvalues of the graphGwhere the eigenvalue of a graphGis the eigenvalue of its adjacency matrixAG. Ultimately, scores of various graph energies have been originated. It has been shown in this paper that the different graph energies of the regular splitting graphS′Gis a multiple of corresponding energy of a given graphG.

A vertex v is a peripheral vertex in G if its eccentricity is equal to its diameter, and periphery $P\left(G\right)$ is a subgraph of G induced by its peripheral vertices. Further, a vertex v in G is a central vertex if $e\left(v\right)=rad\left(G\right)$ , and the subgraph of G induced by its central vertices is called center $C\left(G\right)$ of $G.$ Average eccentricity is the sum of eccentricities of all of the vertices in a graph divided by the total number of vertices, i.e., $avec\left(G\right)=\{\frac{1}{n}\sum e_G\left(u\right);u\in V\left(G\right)\}.$ If every vertex in G is central vertex, then $C\left(G\right)=G$ , and hence, G is self-centered. In this report, we find the center, periphery and average eccentricity for the convex polytopes.

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.

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 $K_{3} \boxtimes P _{n},~n \geq 3$ .

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.