Omega Index of Line and Total Graphs

Musa Demirci,Ismail Naci Cangul,Ahmet Sinan Cevik,Sadik Delen,Elena Guardo

doi:10.1155/2021/5552202

Abstract

A derived graph is a graph obtained from a given graph according to some predetermined rules. Two of the most frequently used derived graphs are the line graph and the total graph. Calculating some properties of a derived graph helps to calculate the same properties of the original graph. For this reason, the relations between a graph and its derived graphs are always welcomed. A recently introduced graph index which also acts as a graph invariant called omega is used to obtain such relations for line and total graphs. As an illustrative exercise, omega values and the number of faces of the line and total graphs of some frequently used graph classes are calculated.

Highlights

Of all vertices where Δ is the biggest vertex degree in G, is called the degree sequence of the graph
The total graph T(G) of G is the graph whose vertex set is V(G) ∪ E(G) with two vertices of T(G) being adjacent iff the corresponding elements of G are either adjacent or incident
Minimal doubly resolving sets and strong metric dimension of the layer sun graph and the line graph of this graph are calculated in [4]. e classical meanness property of some graphs based on line graphs was considered in [5]

Summary

Introduction

Of all vertices where Δ is the biggest vertex degree in G, is called the degree sequence of the graph. A graph which is connected and has no cycles is called a tree. Given a graph G, the line graph L(G) of G is the graph whose vertex set is E(G) with two vertices of L(G) being adjacent iff corresponding edges in G are adjacent. For some applications of the line graph, see, e.g., [4, 5]. The total graph T(G) of G is the graph whose vertex set is V(G) ∪ E(G) with two vertices of T(G) being adjacent iff the corresponding elements of G are either adjacent or incident. For some recent applications of the total graphs, see, e.g., [6,7,8]

Omega Index and Fundamentals

Omega and r of the Line Graphs of Some Special Graphs

Omega and r of the Total Graphs of Some Special Graphs

Conclusion