Total Energy of Cycle and Some Cycle Related Graphs

In this article we write algorithms and MATLAB programs to find the total energy of Cycle and some Cycle related graphs. The concept of total matrix and total energy of a graph G is introduced by K.Palani&M.Lalithakumari in [9]. Let G=(V, E) be a (p, q) simple graph. Let V (G) = {ν i /i = 1,2, … p} and E (G) = {e i /i = 1,2, … q}. The total matrix T = T(G) of G is a square matrix of order p + q whose (i, j)-entry is defined as: T=(tij)={1ifviadjacent tovji≠j1ifeiadjacent toeji≠j1eiincident withvj0otherwise The Total Energy of a graph is the sum of absolute value of the eigen values of its Total matrix T(G). For any (p, q) graph G, the total number of eigen value is p+q. Let λ 1, λ 2, λ 3, … λ p+q be the eigen values of T. Then total energy of G is T E = ∑i+1p+q|λi| .


I. Introduction
Throughout this article we deal with finite, simple and undirected graphs. The concept of energy of a graph was proposed by Gutman [7] in 1978 as the sum of absolute values of the eigen value of a graph G and is denoted by ‫.)ܩ(ܧ‬ The eigen values of the total matrix T is known as the total eigen values of G. We find the total energy for Path, Star, ܻ ݊+1 & Bull Graph.

Definition
The total graphT (G) of a graph G is a graph such that (i) the vertex set of T (G) corresponds to the vertices and edges of G and (ii) two vertices are adjacent in T (G) if and only if their corresponding elements are either adjacent or incident in G.

Definition
Let ‫ܩ‬ = (ܸ, ‫)ܧ‬ be a (p,q) graph. The energy of total matrix of G is called the total energy. It is denoted as TE(G).
That is the total energy of G= Energy of Total matrix of G ‫ݍ+‬ = ∑|ߣ ݅ | ݅=1 II. Total Energy of cycle related graphs graph 2.1 Algorithm to generate the total energy of cycle graph .
Step I: Assume G=(V,E) to be a (p,q) graph.
Step II: Assume that in the total matrix representation, vertices and edges appear alternatively along both rows and columns.
Step VIII: Assume the other entries as zero.
Step IX: Find eigen values of T.
Step X: Find Total Energy TE.

MATLAB program to generate the total energy of cycle graph
% "T" is the Total matrix of a graph % "K" is the eigen values of the matrix % "TE" is the Total Energy of the graph % r=p+q % p,q refers the number of vertices and edges of ‫ܥ‬ ݊ for i=1:r-1

Illustration
When the above program is executed for ‫,6ܥ‬ the output will be TE=18.93

Algorithm to generate the total energy of Dumbbell graph .
Step I: Assume G= (V,E) to be a (p,q) graph.
Step II: Assume that in the total matrix representation, vertices and edges appear alternatively along both rows and columns.
Step XIII: Assume the other entries as zero.
Step XIV: Find eigen values of T.
Step XV: Find Total Energy TE.

Illustration
When the above program is executed for ‫,5ܦ‬ the output will be TE=36.7617

Algorithm to generate the total energy of Pan graph .
Step I: Assume G= (V, E) to be a (p,q) graph.
Step II: Assume that in the total matrix representation, vertices and edges appear alternatively along both rows and columns.
Step IX: Assume the other entries as zero.
Step X: Find eigen values of T.
Step XI: Find Total Energy TE.

MATLAB program to generate the total energy of Pan graph
% "T" is the Total matrix of a graph % "K" is the eigen values of the matrix % "TE" is the Total Energy of the graph % r=p+q % p,q refers the number of vertices and edges of ܲܽ ݊ for i=1:r-1

Illustration
When the above program is executed for ܲܽ5, the output will be TE=20.4252 2.10 Algorithm to generate the total energy of graph < , − >.
Step I: Assume G= (V, E) to be a (p,q) graph.
Step II: Assume that in the total matrix representation, vertices and edges appear alternatively along both rows and columns.
Step XII: Find eigen values of T.
Step XIII: Find Total Energy TE.
2.11 MATLAB program to generate the total energy of graph < , − >.

Illustration
When the above program is executed for < ‫,5ܥ‬ ‫4ܥ‬ >., the output will be TE=30.5443