An Unique and Novel Graph Matrix for Efficient Extraction of Structural Information of Networks

In this article, we propose a new type of square matrix associated with an undirected graph by trading off the naturally imbedded symmetry in them. The proposed matrix is defined using the neighbourhood sets of the vertices. It is called as neighbourhood matrix and it is denoted by $ \mathcal{NM}(G)$ as this proposed matrix also exhibits a bijection between the product of the two graph matrices, namely the adjacency matrix and the graph Laplacian. This matrix can also be obtained by looking at every vertex and the subgraph with vertices from the first two levels in the level decomposition from that vertex. The two levels in the level decomposition of the graph give us more information about the neighbour of a vertex along with the neighbour of neighbour of a vertex. This insight is required and is found useful in studying the impact of broadcasting on social networks, in particular, and complex networks, in general. We establish several interesting properties of the $ \mathcal{NM}(G) $. In addition, we also show how to reconstruct a graph $G$, given a $ \mathcal{NM} (G)$. The proposed matrix is also found to solve many graph theoretic problems using less time complexity in comparison to the existing algorithms.


Introduction
In the study of complex and social networks, one of the interesting and challenging problems is to study the impact of a change that occurs to a node. In the literature, such studies are carried out to analyse the network's behavioural changes both locally as well as globally, [12]. One such problem is to reconstruct a graph when partial information is known and to predict the dynamical changes occurring in a network. To tackle this problem, we were determined to approach it by studying graphs through their matrices.
Matrices play a vital role in the study of graphs and their representations. Among all the graph matrices, adjacency matrix and Laplacian matrix has received extensive attention due to their symmetric nature and the ability to exhibit various properties [2,6,9]. In the literature, many other types of matrices have been associated with a graph [1,3,7,9,10]. The spectral studies on graph matrices have also received extensive attention in the literature [8,11,13]. For an undirected graph, most of the matrices are symmetric and not of help to solve our problem. Further, in [4], the authors discuss the product of two graphs and its representation using the product of the adjacency matrices of the graphs. Also, powers of adjacency matrix and square of distance matrix has also been studied in the literature [3]. However, there is no literature dealing with the product of two types of matrices of a graph.
In this paper, we handle one such problem involved in defining, analysing and correlating the product of graph matrices with the graph and several of its properties. To this end, we propose a novel representative matrix for a graph referred to as N M(G). We first define this matrix by using the notion of the neighbourhood of a vertex in a graph and then endorse its relationship with the product of two different types of graph matrices. We make sure that the matrix that we are defining in this paper is not always symmetric, and this helps us in proving many network properties quite easily.
The organisation of this paper is as follows: In section 2, we present all the basic definitions, notations and properties required. In section 3, we introduce the novel concept of N M(G) and discuss several of its properties. In section 4, we discover some interesting characterisations of the graph using the N M(G). We conclude the paper in section 5 with some insight on the future scope.

Definitions and Notations
Throughout this paper, we consider only undirected, unweighed simple graphs. For all basic notations and definitions of graph theory, we follow the books by J.A. Bondy and U.S.R. Murty [5] and D.B. West [14]. In this section, we present all the required notations and define the N M(G). Let G(V, E) be a graph with vertex set V (G) and edge set E(G).
denote the adjacency matrix of the graph G. Let the degree matrix D(G) (or D ) be the diagonal matrix with the degree of the vertices as its diagonal elements. Let C(G) be the Laplacian matrix obtained by www.ejgta.org A unique and novel graph matrix | S. Karunakaran and L. Selvaganesh Definition 1. Given a graph G, the product of the adjacency matrix and the degree matrix, denoted by AD = [ad ij ], is defined as Similarly, the product of the degree matrix and the adjacency matrix, denoted by DA = [da ij ], is defined as Remark 2.1. From the above definitions it follows immediately that (AD) T = DA.
Remark 2.2. If G is regular or contains regular-components then by the definition, AD matrix is symmetric. Hence by above remark AD and DA becomes equal.
Definition 2. Given a graph G, the square of the adjacency matrix A 2 = [a 2 ij ], is defined as It is well known that the ij th entries of the square of adjacency matrix denotes the number of walks of length 2 between i and j.
We now extend the above notion of product of graph matrices to obtain a new class of matrix and establish its properties.

N M(G) and its properties
Now we introduce the idea of N M(G) and describe its properties Definition 3. Given a graph G, the neighbourhood matrix, denoted by N M(G) = [nm ij ] is defined as    Proof. Consider the definition of product of two matrices Note that the last equality represents the N M(G). Hence the proof. Proof. By Proposition 3.1, it is immediate that the matrix N M(G) can be constructed from the adjacency matrix. Figure 1(b), constructing the adjacency matrix as defined in the above proposition, we get the matrix A as shown in Figure 2. It is immediate that A is the required adjacency matrix.
An alternative interpretation or a way of defining the N M(G) is to consider the breadth first traversal starting at a vertex i. By inspection of the first two levels in this level decomposition, we can obtain the respective i th row of the N M(G). We prove this equivalence in the following proposition.
www.ejgta.org A unique and novel graph matrix | S. Karunakaran and L. Selvaganesh www.ejgta.org Remark 3.2. For an undirected simple graph G, we have Proposition 3.4. The N M(G) matrix for any graph G is a singular matrix.
Proof. Let A be the adjacency matrix and C be the Laplacian matrix of a graph G. It is enough to prove det(N M) = 0. Since Since it is well know that det(C) = 0 we get the last equality and hence the claim.
Consider the level decomposition of the graph G from the vertex i. Observe that, is the number of edges connecting the vertices from level 1 to level 2. Similarly, |N G (i) ∩ N G (j)| denote the number of edges connecting the vertices from level 2 to level 1. So, we have Substitute equation (3) in equation (2) we get the row sum of N M(G) is zero.   In addition, using the row entries and the two level decomposition, we can construct the induced subgraph rooted at vertex i. Here, observe that nm 51 is −2 implies that there are two paths between vertices 5 and 1 of length 2. Similar observation leads to the fact that among the vertices {2, 4, 6}, vertex 1 is adjacent to two of them while vertex 3 is adjacent to the remaining one from the same set. To determine the adjacency of the vertex 1 and vertex 3, we trace the corresponding rows in the N M matrix, namely nm 12 , nm 14 , nm 16 . Figure 3(b) shows the constructed subgraph rooted at vertex 5 by using the corresponding row entries.
Hence the proof.

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A unique and novel graph matrix | S. Karunakaran and L. Selvaganesh

Graph characterization using neighbourhood matrix N M(G)
Note that the matrix N M(G) is not always symmetric. The next result characterizes the graphs for which N M(G) will be symmetric. Proof. Let G be a graph with w(G) components, say G 1 , G 2 , . . . , G w such that each G z is regular with degree r z , 1 ≤ z ≤ w(G). By the definition of N M(G) when i is not adjacent to j then nm ij = nm ji and when i is adjacent to j, then (i, j) ∈ E(G z ), for some z, 1 ≤ z ≤ w(G), and From (4) and (5) we have nm ij = nm ji . Therefore the N M(G) is symmetric when the graph G has regular components. Conversely, let N M(G) be symmetric. We know that, N M(G) can be written as AD − A 2 . Since sum of symmetric matrices is symmetric and AD = N M + A 2 , we must have AD to be symmetric. But from Remark 2.2, it is known that AD is symmetric whenever G is the union of regular components.
Recall that a graph G is said to be a strongly regular graph with parameters (n, k, µ 1 , µ 2 ), if G is a k-regular graph on n vertices in which every pair of adjacent vertices has µ 1 common neighbours and every pair of non-adjacent vertices has µ 2 common neighbours. Proof. By the definition of N M(G) it immediate follows that for a strongly regular graph G, where . This implies the entries of N M(G) of a strongly regular graph takes values from   Figure 4(b) is the corresponding graph of Figure 4(a). Note that the graph is not a strongly regular graph.  Proof. Let i th row of N M(G) have no zero entries then by using the two level decomposition   Proposition 4.6. Given a graph G, the number of triangles in G is given by Proof. Given a vertex i, when i is adjacent to j and there exists at least one common neighbour x, for i and j, we get a triangle. Therefore, the number of triangles containing the vertex i is given by , since a triangle < i, j, x, i > will be counted twice, one for each j, x ∈ N G (i). Hence, Total number of triangles in the graph = 1 3 Hence the claim.
Proof. Given a graph G on n vertices, the number of 4-cycles containing the vertex i is given by Hence the total number of 4-cycles (both induced and not induced) can be given by, Remark 4.4. Note that in the above proof, . This implies that G has no induced C 4 .
Recall that the girth of a graph is the length of a shortest cycle contained in the graph.
Proposition 4.9. A graph G has girth at least 5 if and only if nm ij = |nm jj | for every edge (i, j) ∈ E(G) and nm ij ≥ −1, for every pair (i, j) / ∈ E(G).
Proof. By Proposition 4.5, we have that the graph G is Triangle free if and only if nm ij = |nm jj |, for every edge (i, j) ∈ E(G). Also, by Proposition 4.8 we have that the graph has no induced C 4 if and only if for every pair (i, j) / ∈ E(G), nm ij ≥ −1. Therefore we can conclude G has girth at least 5.

Conclusion and Future directions
In this paper, we have introduced a new graph matrix (N M(G)) that can be associated with a graph to reveal more information when compared to the adjacency matrix. We have also systematically demonstrated the equivalence of the N M(G) and the product of two other existing graph matrices, namely adjacency and Laplacian matrices. Further, we have endorsed its relationship with the concept of level decomposition of the graph. Further, we have also substantiated the usefulness of the N M(G) by identifying numerous properties with the aid of this matrix. In this process, we have shown many simple properties, such as counting the number of triangles in a graph, can be done in minimal time.
In our first attempt to analyse a new graph matrix, we have only studied its correctness and a few of its properties in this paper. This graph matrix seems to be quite promising and be applicable A unique and novel graph matrix | S. Karunakaran and L. Selvaganesh in studying problems relating to domination in graphs and graph isomorphism problem. As an extension of this current work, our subsequent research article comprises of the study of the N M spectrum.