Energy of Certain Classes of Graphs Determined by Their Laplacian Degree Product Adjacency Spectrum

)e graph energy was firstly introduced by Ivan Gutman in 1978 [1]. His idea was motivated by the well-known Hückel molecular orbital theory by Erich Hückel in 1930s, which permits pharmacologists to imprecise energies associated π-electron orbital of molecules called conjugated hydrocarbons [2]. )e spectrum and the energy of a graph have significant applications and connections in the branches of Mathematics, such as linear algebra and combinatorial optimization fields which have lot to do with graph spectrum and energy. )e combinatorial and graph theoretical approaches have strong bonding to solve real-life problems. Many results and methods from the spectral graph theory can be applied for the practicalities and evolution of matrix theory [3]. An ordered pair Γ � (V, E), called a graph with vertex set of Γ, is denoted by V and its edge set by E. Two vertices u, v are adjacent if they make an edge in Γ, and we denote it by u ∼ v.)e number of edges incident to a vertex v of Γ is the degree of v, and it is denoted by d(v) [4, 5]. )e adjacencymatrix of Γ, of order n denoted byA(Γ), is a square symmetric matrix of order n × n whose ijth element can be found as [3] aij � 0, if u≁v,


Introduction
e graph energy was firstly introduced by Ivan Gutman in 1978 [1]. His idea was motivated by the well-known Hückel molecular orbital theory by Erich Hückel in 1930s, which permits pharmacologists to imprecise energies associated π-electron orbital of molecules called conjugated hydrocarbons [2]. e spectrum and the energy of a graph have significant applications and connections in the branches of Mathematics, such as linear algebra and combinatorial optimization fields which have lot to do with graph spectrum and energy. e combinatorial and graph theoretical approaches have strong bonding to solve real-life problems. Many results and methods from the spectral graph theory can be applied for the practicalities and evolution of matrix theory [3]. An ordered pair Γ � (V, E), called a graph with vertex set of Γ, is denoted by V and its edge set by E. Two vertices u, v are adjacent if they make an edge in Γ, and we denote it by u ∼ v. e number of edges incident to a vertex v of Γ is the degree of v, and it is denoted by d(v) [4,5]. e adjacency matrix of Γ, of order n denoted by A(Γ), is a square symmetric matrix of order n × n whose ijth element can be found as [3] a ij � 0, if u≁v, For energy and spectrum of graph Γ, let A(Γ) be the adjacency matrix, the summation of absolute values of its eigenvalues compose energy of graph and these eigenvalues related with their multiplicities forms the spectrum of graph [4], i.e., and where n(λ 1 ), n(λ 2 ), . . . , n(λ n ) are the multiplicities of the eigenvalues λ 1 , λ 2 , . . . , λ n of A(Γ). In [6], the degree product adjacency matrix, for a simple connected graph Γ having n vertices say v 1 , v 2 , . . . , v n , is a real symmetric matrix, denoted by DP e Laplacian degree product adjacency matrix of Γ is defined as where D(Γ) is the degree matrix of Γ having diagonal entries as the degree of each vertex and all other entries are zero. e spectrum (1) and energy (2) obtained correspond to the eigenvalues of L DP A (Γ) and are called the Laplacian degree product adjacency spectrum and energy, LSp DP A (Γ) and LE DP A (Γ), respectively [7], as the degree sum concept was conceived earlier in [8].

Main Results
In this module, we study the Laplacian degree product adjacency spectrum and energy of some well-known families of graphs, such as complete graphs, complete bipartite graphs, friendship graphs, and corona products of 3 and 4 cycles with null graph. We also evaluate the correct spectrum and the energy of degree product adjacency matrix of the corona product of 4 cycle with null graphs (thorny 4-cycle ring), which was found incorrect in [6].

Complete Graphs
, v x be the vertex set of K x ; then, the following result provides the Laplacian degree product adjacency spectrum and energy of K x .
Theorem 1. For x ≥ 2, let K x be a complete graph. en, and Laplacian degree product adjacency energy of Accordingly, we have the following Laplacian degree product adjacency matrix: Eigenvalues of L DP A (K x ) are ese eigenvalues provide the required spectrum. Moreover, by (3), we have Since the size of K x is x 2 , so the result is proved. □

Complete Bipartite Graphs
. . , v y be as partitions. e order and the size of K x,y graph are x + y and xy, respectively. en, the Laplacian degree product adjacency spectrum and energy of K x,y can be obtained from the following result.
Theorem 2. For x, y ≥ 1, a complete bipartite graph K x,y , then Moreover, Accordingly, the Laplacian degree product adjacency matrix of K x,y is Next, we have four cases to discuss.
e required spectrum can be obtained by these eigenvalues. Furthermore, by (3), we have Journal of Mathematics Case II (y ≠ x + 1 with y > x ≥ 1): we get the following eigenvalues of L DP A (K x,y ): (15) e required spectrum can be obtained by these eigenvalues. Moreover, by (3), we have Case III (x � y ≥ 1): eigenvalues of L DP A (K x,x ) are as follows: ese eigenvalues provide the required spectrum. Furthermore, by (3), we have Case IV (x � 1 and y ≥ 1): we get eigenvalues of L DP A (K 1,y ) as follows: ese eigenvalues provide the required spectrum. Using (3), we have the following energy of K 1,y : It completes the proof.

Friendship Graphs F x .
A friendship graph F x has 2x + 1 vertices, and it can be assembled by connecting x clones of the cycle C 3 with a common vertex. Let the vertex set of ith Theorem 3. For x ≥ 2, let a friendship graph F x ; then, en, the Laplacian degree product adjacency matrix is as follows: e eigenvalues of Laplacian degree product adjacency matrix of F x are 2, (x − 1) − times, (24) e required spectrum can be obtained by these eigenvalues. ese eigenvalues provide the following energy:

Corona Products of 3 and 4 Cycles with Null Graphs.
e corona product of graphs Γ and Ω is expressed as Γ°Ω . It can be made by drawing one copy of Γ and |V(Γ) | copies of Ω and connecting the ith vertex of Γ with each vertex of ith copy of Ω [9][10][11]. Let Γ be an x-cycle C x with vertices v 1 , v 2 , . . . , v x and Ω be a null graph N k . en, the vertex set of C x°Nk is where the set v j i ; 1 ≤ i ≤ k is the vertex set of jth copy of N k in C x°Nk . In this portion, we evaluate the Laplacian degree product spectrum and energy of C x°Nk for x � 3 and 4.

Theorem 4.
For k ≥ 1, let the corona product be C x°Nk ; then, Proof. Note that d(v j ) � k + 2, for each j � 1, 2, 3, and d(v j i ) � 1, for each 1 ≤ j ≤ 3 and 1 ≤ i ≤ k. For the convenience, we let k + 2 � α. en, the Laplacian degree product e eigenvalues obtained from the above matrix of C x°Nk are en, the required spectrum can be obtained by these eigenvalues. Also, by (3) Proof. Note that d(v j ) � k + 2, for eachj � 1, 2, 3, 4, and d(v j i ) � 1, for each 1 ≤ j ≤ 4 and 1 ≤ i ≤ k. For the convenience, we let k + 2 � α. en, the Laplacian degree product adjacency matrix of C 4°Nk is as