Certain Energies of Graphs for Dutch Windmill and Double-Wheel Graphs

Energy of a graph is defined as the sum of the absolute values of the eigenvalues of the adjacency matrix associated with the graph. In this research work, we find color energy, distance energy, Laplacian energy, and Seidel energy for the Dutch windmill graph of cycle lengths 4, 5, and 6. Also, we find the lower bounds of the double-wheel graph for energy, Seidel energy, color energy, distance energy, Laplacian energy, and Harary energy.


Introduction
Energy of a graph is one of the important concepts in graph theory. e energy of a graph is mostly studied in the context of spectral graph theory. Gutman [1] introduced the concept of energy of a graph for the first time. Gutman defined energy of a graph as the sum of the absolute values of the eigenvalues of the adjacency matrix associated with the graph. Adjacency matrix is used to represent a finite graph, and it is a square matrix. e entries of the adjacency matrix provide information whether any two vertices are joined with each other or not. e total pi electron energy of conjugated hydrocarbons in chemistry computed by using Huckel molecular orbital theory coincides with the energy defined by Gutman, so the energies calculated in graph theory have a special significance.
Calculating the bounds for graph spectra has been the main area of research among graph theorists. McClelland [2] worked on the estimation of pi electron energies. Nageswari and Sarasija [3] calculated edge energy bounds of finite, simple, and undirected graphs. Das and Gutman [4] calculated some upper and lower bounds for E(G) in terms of number of vertices and number of edges. Adiga and Rakshith [5] calculated upper bounds for the extended energy of graphs. Jahanbani [6] calculated lower bounds for the energy of graphs. In this research work, we calculate lower bounds for various energies of the double-wheel graph for the first time. Now, we introduce some terminologies associated with our work.
In mathematics, in graph theory, we study mathematical structures which are called graphs, and we use these graphs to represent pairwise relations between different objects. So, graph theory is the study of graphs and their characteristics. One of the most important concepts in graph theory is energy of a graph. is concept was introduced by Gutman [1] which defined energy as the sum of the absolute values of the roots of the characteristics equation, and these roots are called eigenvalues of the graph under consideration.
In [7], energy, Seidel energy, Harary energy, distance energy, color energy, and Laplacian energy of the friendship graph have been calculated. In this research paper, we calculate these energies of the Dutch windmill graph of cycle lengths 4, 5, and 6, and also, we find lower bounds of these energies of the double-wheel graph.

Basic Definitions and Notations
Definition 1. For a graph G, the adjacency matrix is the square matrix of order n × n denoted by A(G) and is defined as e adjacency matrix gives us information whether a vertex in a pair of vertices is joined with the other vertex or not.

Definition 2.
e Laplacian matrix is also a square matrix defined as follows: where DE G(G) is the diagonal matrix of vertex degrees and A(G) is the adjacency matrix of graph G.
Definition 3. e Harary matrix of a graph G is the square matrix of order n whose (i, j) entry is defined as (1/d ij ) where d ij is the distance between the vertices v i and v j . Definition 5. e distance matrix of G is a matrix of order n, and its entries represent shortest distances between its different vertices.

Definition 6.
e Seidel matrix of a graph is denoted by e Seidel matrix is a square, real symmetric matrix of order n. e eigenvalues of the Seidel matrix are the eigenvalues of the graph G.

Definition 7.
e sum of the absolute values of the roots of the characteristic equation associated with the adjacency matrix of the graph determines the energy of that graph. If λ 1 , λ 2 , λ 3 , . . . , λ n are the eigenvalues of the adjacency matrix, then energy of graph E(G) is defined as follows: Haemers [7] introduced Seidel energy of a graph. To calculate the Seidel energy of a graph, we make its Seidel matrix as defined in equation (4) and find its characteristic equation; now, the sum of the absolute values of the roots of this characteristic equation determines the Seidel energy of the graph under consideration.

Theorem 3.
e energy of a Dutch windmill graph D m 6 is Proof. e adjacency matrix for D m 6 is e eigenvalues of the above characteristic equation are /2). Now, energy can be calculated by using Definition 2.4.1 as follows: 4 Journal of Mathematics Proof. e adjacency matrix of order 2n + 1 for a doublewheel graph W n , n ≥ 3, is e characteristic equation of the above adjacency matrix is e three roots of the above equation are λ 1 � 2, λ 2 � 1 + ����� 1 + 2n √ , and λ 3 � 1 − ����� 1 + 2n √ . Now, the energy of the double-wheel graph can be calculated by using Definition 2.4.1 as follows: Proof. e Seidel matrix for D m 4 is

Journal of Mathematics
e characteristic equation of the above Seidel matrix is (m − 1times), and λ 4 , λ 5 , λ 6 , where λ 4 , λ 5 , λ 6 are the roots of the cubic equation given in the following: Now, Seidel energy can be calculated by using Definition 2.4.2 as follows: is given by Proof. e Seidel matrix for D m 5 is e characteristic equation of the above Seidel matrix is e eigenvalues of the above characteristic equation are (m − 1)times, and λ 5 , λ 6 , λ 7 , where λ 5 , λ 6 , λ 7 are the roots of the cubic equation given in the following: Now, Seidel energy can be calculated by using Definition 2.4.2 as follows: □ Journal of Mathematics 7 e characteristic equation for the above Seidel matrix is e eigenvalues of the above Seidel matrix are (m − 1times), and λ 5 , λ 6 , λ 7 , λ 8 where λ 5 , λ 6 , λ 7 , λ 8 are roots of the biquadratic equation given in the following: Now, Seidel energy can be calculated by using Definition 2.4.2 as follows: e characteristic equation of the Seidel matrix is /2). Now, the Seidel energy can be calculated by using Definition 4 as follows: Proof. e Laplacian matrix for D m 4 is given as follows: e characteristic equation of the above Laplacian matrix is given as follows: Eigenvalues are λ 1 � 0, λ 2 � 2(m times), (39) e characteristic equation of the above Laplacian matrix is given as follows: (40) e eigenvalues of the above Laplacian matrix are given as follows.
Proof. e color matrix for D m 4 is given as follows: Journal of Mathematics 13 e characteristic equation of the above color matrix is given as follows: (m − 1 times), and λ 4 , λ 5 , λ 6 are the eigenvalues of the above characteristic equation, where λ 4 , λ 5 , λ 6 are the roots of the cubic equation given in the following: So, the color energy of where χ � |λ 4 | + |λ 5 | + |λ 6 |.