On the Vector Degree Matrix of a Connected Graph

A matrix representation of the graph is one of the tools to study the algebraic structure and properties of a graph. In this paper, by deﬁning the vector degree matrix of graph G , we provide a new matrix representation of the graph. For some standard graphs, VD-eigenvalues, VD-spectrum, and VD-energy values are deﬁned and calculated. Moreover, we calculate the VD-matrix and calculate the VD-eigenvalues for graphs representing the chemical composition of paracetamol and tramadol.


Introduction
e relationship between structural and spectral properties of a network could be a key question within the field of Network Science. Spectral graph strategies have become an elementary tool in the analysis of huge advanced networks and connected disciplines, with a broad variety of applications in machine learning, data processing, Internet search and ranking, scientific computing, and PC vision. e central topic of pure mathematics graph theory is learning the relationship between a graph's structure and its eigenvalues. Specifically, the eigenvalues of matrices representing the graph structure, like the contiguity or the Laplacian matrices, have an on-the-spot associated with the behavior of many networked slashing processes, like spreading processes, synchronization of oscillators, random walks, accord dynamics, and a large type of distributed algorithms. Details on spectra and the theory of graph energy can be found in [1,2,[4][5][6][10][11][12][13][14][15], whereas details on its chemical applications are found in the book [9] and the review [7,8].
Given a graph, we can associate several matrices that record information about vertices and how they are interconnected. In this paper, we introduced a new matrix based on the distances between the vertices and their degrees. By calculating the new eigenvalues for some standard graphs, we study some properties of this new matrix.

The Vector Degree Spectrum of a Graph
e characteristic polynomial det(cI − VD(G)) of VD(G) is called the VD-characteristic polynomial of G and is denoted by P VD (G) � n i�0 a i c n− i . e eigenvalues of the matrix VD(G) which are the zeros of |cI − VD(G)| are called the VD-eigenvalues of G and form its Spectrum denoted by Spec VD (G). If the distinct VD-eigenvalues of G are c 1 , c 2 , . . . , c m with multiplicities t 1 , t 2 , . . . , t m , respectively, then Spec VD (G) is written as By the above definition, the degree matrix is a real symmetric n × n matrix. erefore, its eigenvalues c 1 , c 2 , . . . , c m are real numbers. Since the trace of VD(G) is zero, the sum of its eigenvalues is also equal to zero.

Lemma 2.
Let G be a connected graph with n vertices, and let c 1 , c 2 , . . . , c n be its VD-eigenvalues. en, Definition 2. e Vector degree energy of the graph G is Theorem 1. For the complete graph K n of order n ≥ 2, Proof. Let G � K n be a complete graph with vertices v 1 , v 2 , . . . , v n . en, for every vertex v i , i � 1, 2, . . . , n, the vector of a degree corresponding to the vertex v i → � (1, n − 1). us, for any two vectors v i → and v j → , Hence, is the adjacency matrix of K n . erefore, Lemma 1 gives that □ We now determine the VD-spectrum and VD-energy of any cycle C n . Theorem 2. Let n ≥ 4 be an even integer. en, for a cycle C n , we have Further, Proof. By labeling the vertices of the cycle C n in the anticlockwise direction as v 1 , v 2 , . . . , v n , we observe that for en, clearly for any two vertices v i and v j , erefore, VD(C n ) � 2(n − 1)A(K n ). Hence, by Lemma 1, we get and it is easy to see that where E(K n ) is the energy of K n . □ Theorem 3. Let n ≥ 3 be an odd integer. en, for a cycle C n , we have Further Proof. By labeling the vertices of the cycle C n in the anticlockwise direction as v 1 , v 2 , . . . , v n , we observe that for en, clearly for every two vertices erefore, VD(C n ) � (2n − 1)A(K n ). Hence, by Lemma 1, we get, and r 1 and r 2 are the zeros of the polynomial. Furthermore, Put α � 1 + m 2 + (n − 1) 2 , β � 1 + nm + (n − 1)(m − 1), and c � 1 + n 2 + (m − 1) 2 . erefore, where A � αA(K n ), B is an n × m matrix with all entries β, and C � cA(K m ). eorem 1 gives and the corresponding basis of the eigen space E − α is us, − α is an eigen value of VD(K n,m ), and the corresponding basis of the eigen space E − α ≤ R n+m is Similarly, − c is an eigen value of VD(K n,m ) with multiplicity m − 1. Consequently, the characteristic polynomial can be written as (x + α) n− 1 (x + c) m− 1 p(x) where p(x) is a polynomial of degree 2 and by a routine calculations, where r 1 and r 2 are the zeros of the polynomial. Furthermore, E D (K n,n ) � (3n − 2)(n 2 − 2n + 2).

Corollary 2.
Let G be any star graph K 1,n . en, where r 1 and r 2 are the zeros of the polynomial:

Vector Degree Spectrum and Energy of Regular and Strongly Regular Graphs
One of the most important families of regular graphs is strongly regular graphs (abbreviated SRG), which has so many beautiful properties. ere are many SRGs which arise from combinatorial concepts such as orthogonal arrays, Latin squares, conference matrices, designs, and geometric graphs. A strongly regular graph (SRG) with parameters (n, k, λ, μ) is a graph on n vertices which is regular with valency k and has the following properties: (i) Any two adjacent vertices have exactly λ common neighbors (ii) Any two nonadjacent vertices have exactly μ common neighbors [3] Theorem 5 (see [5]). Let G be a strongly regular graph with parameters (n, k, λ, μ). en, the eigenvalues of G satisfy the following properties: (1) G has exactly three distinct eigenvalues which are k, θ, and τ where (2) e multiplicity of the eigenvalue k is 1, and the multiplicities of θ and τ are f and g, respectively, where (3) If (n − 1)(μ − λ) − 2k ≠ 0, then the eigenvalues θ and τ are integers. On the other hand, if (n − 1)(μ − λ) − 2k � 0, then f � g and θ and τ need not be integers. e strongly regular graph is called a conference graph in this case.

Theorem 6.
Let G be a strongly regular graph with parameters (n, k, λ, μ). en, and Proof. Let G be a strongly regular graph with the parameters (n, k, λ, μ). For any vertex v clearly v → � (1, k, n − k − 1), then the degree product of any two-degree vectors is equal to 1 + k 2 + (n − k − 1) 2 . Hence, If we put δ � (1 + k 2 + (n − k − 1) 2 ), then and E VD (G) � δE(K n ) We can generalize eorem 6 as the follows. □ Theorem 7. Let G � (V, E) be a k-regular graph of diameter two. en, and In the following, we calculate VD-matrix and calculate VD-eigenvalues for graphs representing the chemical composition of paracetamol and tramadol. Also, we note that the closeness of eigen values between paracetamol and tramadol might indicate pharmacological and chemical characteristics convergence. 4 Journal of Mathematics Scheme 1: e graph of Paracetamol.

Journal of Mathematics
Scheme 2: e graph of Tramadol.