An Unique and Novel Graph Matrix for Efficient Extraction of Structural Information of Networks
aa r X i v : . [ c s . D M ] M a r An Unique and Novel Graph Matrix for Efficient Extraction of StructuralInformation of Networks
Sivakumar Karunakaran a , Lavanya Selvaganesh b, ∗ a SRM Research Institute, S R M Institute of Science and Technology Kattankulathur, Chennai - 603203, INDIA b Department of Mathematical Sciences, Indian Institute of Technology (BHU), Varanasi-221005, INDIA
Abstract
In this article, we propose a new type of square matrix associated with an undirected graph by trading offthe naturally imbedded symmetry in them. The proposed matrix is defined using the neighbourhood sets ofthe vertices. It is called as neighbourhood matrix and it is denoted by
N M ( G ) as this proposed matrix alsoexhibits a bijection between the product of the two graph matrices, namely the adjacency matrix and thegraph Laplacian. This matrix can also be obtained by looking at every vertex and the subgraph with verticesfrom the first two levels in the level decomposition from that vertex. The two levels in the level decompositionof the graph gives us more information about the neighbour of a vertex along with the neighbour of neighbourof a vertex. This insight is required and is found useful in studying the impact of broadcasting in socialnetworks, in particular, and complex networks, in general. We establish several interesting properties of the N M ( G ). In addition, we also show how to reconstruct a graph G , given an N M ( G ). The proposed matrixis also found to solve many graph theoretic problems using less time complexity in comparison to the existingalgorithms. Keywords:
Graph Matrices, Graph Characterization, Product of Matrices, Graph Properties.Mathematics Subject Classification :
MSC , 05C50, 05C62, 05C82
1. Introduction
In the study of complex and social networks, one of the interesting and challenging problem is to study theimpact of a change that occurs to a node. Such studies are being done to analyse the network’s behaviouralchanges both locally as well as globally, [6]. One such problem is in reconstructing a graph when partialinformation is known and/or 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 an important role in the study of graphs and their representations. It is well known thatfor undirected graphs, among all graph matrix forms, adjacency matrix and Laplacian matrix has receivedwide attention due to their symmetric nature [1, 3, 4]. In the literature, many other types of matrices thatcould be associated with a graph [1, 2, 4, 5]. For an undirected graph, every such matrix is found to besymmetric and is not of help to solve our problem. Further, in [2], the authors discuss about the product oftwo graphs and its representation using product of the adjacency matrices of the graphs. However, there isno 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 ofgraph matrices with the graph and several of its properties. To this end, we propose a novel representative ∗ Corresponding author.
Email addresses: [email protected] (Sivakumar Karunakaran), [email protected] (Lavanya Selvaganesh) A preliminary version of this article namely the definition of the newly proposed matrix was presented in ICDM 2016(June09-11), Siddaganga Institute of Technology, Tumkur-572102, Karnataka, INDIA and few of the characterizations were presentedin the Fifth India-Taiwan Conference on Discrete Mathematics(18-21 July, 2017), Tamkang University, Taiwan. atrix for a graph referred to as
N M ( G ). We first define this matrix by using the notion of neighbourhoodof a vertex in a graph and then endorse its relationship with the product of two different types of graphmatrices. We make sure that the matrix that we are defining in this paper is not always symmetric and thishelps us in proving many network properties quite easily.The paper is organized as follows: In section 2, we present all the basic definitions, notations andproperties required. In subsection of 2, we introduce the novel concept of N M ( G ) and discuss several of itsproperties. In section 3, we discover some interesting characterizations of the graph using the N M ( G ). Weconclude the paper in section 4 with some insight on future scope.
2. Definitions and Notations:
Throughout this paper, we consider only undirected, unweighed simple graphs. For all basic notationsand definitions of graph theory, we follow the books by J.A. Bondy and U.S.R. Murty [3] and D.B. West [7].In this section, we present all the required notations and define the
N M ( G ). Let G ( V, E ) be a graph withvertex set V ( G ) and edge set E ( G ). For a vertex v ∈ V ( G ), let N G ( v ) denote the set of all neighbours of vand N G [ v ] = { v } ∪ N G ( v ), denote the closed neighbourhood of v . The degree of a vertex v is given by deg ( v )or | N G ( v ) | . Let A G (or A ) denote the adjacency matrix of the grpah 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 Laplacianmatrix obtained by C ( G ) = D ( G ) − A G . Definition 1.
Given a graph G , the product of the adjacency matrix and the degree matrix, denoted by AD = [ ad ij ] , is defined as ad ij = ( | N G ( j ) | , if ( i, j ) ∈ E ( G )0 , otherwise Similarly, the product of the degree matrix and the adjacency matrix, denoted by DA = [ da ij ] , is defined as da ij = ( | N G ( i ) | , if ( i, j ) ∈ E ( G )0 , otherwise Remark . From the above definitions it follows immediately that AD T = DA.
Remark . 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 = [ a ij ] , is defined as a ij = ( | N G ( i ) | , if i = j | N G ( i ) ∩ N G ( j ) | , if i = j It is well known that the ij th entries of the square of adjacency matrix denotes the number of walks oflength 2 between i and j .Another concept which we require before proceeding to the main result is the level decomposition of agraph with respect to a source node, which is defined by the Breadth First Search Traversal technique. Breadth First Search (BFS) is a graph traversal technique [3] where a node (source node) and its neigh-bours are visited first and then the neighbours of neighbours. The algorithms returns not only a search treerooted at the source node but also a function l : V → N , which records the level of each vertex in the tree,that is, the distance of each vertex from the source node. In simple terms, the BFS algorithm traverses levelwise from the source. First it traverses level 1 nodes (direct neighbours of source node) and then level 2nodes (neighbours of neighbours of source node) and so on. We refer to such a level representation withreference to a source node as the level decomposition from the source node.We next extend the above notion of product of graph matrices to obtain a new class of matrix andestablish its properties. 2 .1. 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 ) = [ η ij ] is defined as η ij = −| N G ( i ) | , if i = j | N G ( j ) − N G ( i ) | , if ( i, j ) ∈ E ( G ) −| N G ( i ) ∩ N G ( j ) | , if ( i, j ) / ∈ E ( G ) Example . A graph G and its corresponding N M ( G ) representation are given in Figure 1. In thisexample, the neighbourhood set of each vertex of G is given by N G (1)= { } , N G (2)= { } , N G (3)= { } , N G (4)= { } , N G (5)= { } , N G (6)= { } , N G (7)= { } .1 2 3456 7 (a) a graph G N M ( G ) = [ η ij ] = − − − − − − −
10 0 − − − − − − − − − − − − − − − − (b) N M corresponding to G Figure 1: A graph G and its N M ( G ). Proposition 2.1.
The
N M ( G ) can also be defined by using the product of adjacency matrix and Laplacianmatrix of a graph G .Proof. Consider the definition of product of two matrices A × C ( G ) = A × ( D ( G ) − A )= AD − A = [ ad ij ] − [ a ij ]= − | N G ( i ) | , if i = j | N G ( j ) − N G ( i ) | , if ( i, j ) ∈ E ( G )0 − | N G ( i ) ∩ N G ( j ) | , if ( i, j ) / ∈ E ( G )Note that the last equality represents the N M ( G ). Hence the proof. Proposition 2.2.
Given a graph G , the N M ( G ) can be obtained from adjacency matrix and vice versa.Proof. By Proposition 2.1, it is immediate that the matrix
N M ( G ) can be constructed from the adjacencymatrix.Given the N M ( G ), if i = j, η ij > η ij = | N G ( j ) − N G ( i ) | for ( i, j ) ∈ E ( G )and if η ij ≤ i = j or ( i, j ) / ∈ E ( G ).Therefore, we can now define a ij = ( , if η ij > , Otherwise Example . From the
N M ( G ) in Figure 1(b), constructing the adjacency matrix as defined in the aboveproposition, we get,It is immediate that A is the required adjacency matrix.3 = [ a ij ] = Figure 2: Adjacency matrix of G constructed from N M ( G ) An alternative interpretation or a way of defining the
N M ( G ) is to consider the breadth first traversalstarting at a vertex i . By inspection of the first two levels in this level decomposition, we can obtain therespective i th row of the N M ( G ). We prove this equivalence in the following proposition. Proposition 2.3.
Given a graph G , the entries of any row of an N M ( G ) corresponds to the subgraph withvertices from the first two levels of level decomposition of the graph rooted at the given vertex with edgesconnecting the vertices in different levels.Proof. Consider any i th row of the N M ( G ). By the definition of N M ( G ), vertex i is adjacent to a vertex j ⇐⇒ η ij >
0. This gives us the neighbours of i , namely N G ( i ), or the first level of the level decomposition.From the following observations, we obtain the vertices that lie in the next level.1. The diagonal entries are always negative and in particular, if η ii = − c , then the degree of the vertexis c and that there will be exactly c positive entries in that row.2. For some positive integer c , if η ij = c then j ∈ N G ( i ) and that there exists c − i and at distance 2 from i through j .3. If η ij = − c , then the vertex j belongs to the second level of the decomposition and moreover, thereexists c paths of length two from vertex i to j . In other words, there exist c common neighboursbetween vertex i and j .4. If an entry, η ij = 0 then the distance between vertex i and j is at least 3 or the vertex j is isolatedCombining these observations, one can easily obtain the subgraph with vertices from the first two levels ofdecomposition of G rooted at the vertex i .On the other hand, from the Breadth first traversal tree rooted at a vertex i and the definition of N M ( G )we can immediately write the corresponding i th row entry by examining the vertices and their position inthe first two levels.Analogous to N M ( G ) we can also define the product matrix MN ( G ) as follows. Definition 4.
The product of Laplacian matrix and adjacency matrix denoted by MN = [ η ′ ij ] is defined as η ′ ij = −| N G ( i ) | , if i = j | N G ( i ) − N G ( j ) | , if ( i, j ) ∈ E ( G ) −| N G ( i ) ∩ N G ( j ) | , if ( i, j ) / ∈ E ( G ) Remark . Note that MN ( G ) can be obtained by C × A = DA − A . Remark . For an undirected simple graph(
N M ) ′ = ( A × C ) ′ = C ′ × A ′ = C × A = MN roposition 2.4. The
N M matrix is a singular matrixProof.
Let A be an adjacency matrix and C ( G ) be the Laplacian matrix.It is enough to prove det ( N M ) = 0. Since det ( N M ) = det ( A × C )= det ( A ) × det ( C )= 0 . Since it is well know that, det ( C ) = 0 we get the last equality and hence the claim. Proposition 2.5.
Row sum of
N M ( G ) is zero.Proof. Consider any i th row in N M ( G ) n X j =1 η ij = X j ∈ N G ( i ) | N G ( j ) − N G ( i ) | − | N G ( i ) | − X j / ∈ N G [ i ] | N G ( i ) ∩ N G ( j ) | (1) n X j =1 η ij = X j ∈ N G ( i ) ( | N G ( j ) − N G ( i ) | − − X j / ∈ N G [ i ] | N G ( i ) ∩ N G ( j ) | (2)Consider the level decomposition of the graph G from the vertex i .Observe that, P j ∈ N G ( i ) ( | N G ( j ) − N G ( i ) | −
1) is the number of edges connecting the vertices from level 1to level 2. Similarly, P j / ∈ N G [ i ] | N G ( i ) ∩ N G ( j ) | denote the edges connecting the vertices from level 2 to level 1.So, we have X j ∈ N G ( i ) ( | N G ( j ) − N G ( i ) | −
1) = X j / ∈ N G [ i ] | N G ( i ) ∩ N G ( j ) | (3)Substitute the equation (3) in equation (2) we get the row sum of N M ( G ) is zero. Remark . Suppose any row of
N M ( G ) is given, the degree of the vertex the row represents can beobtained from the minimum value of that row. By considering this position as the diagonal position of therow (since η ii = −| N G ( i ) | ), hence enables us to identify the vertex that it represent. Example . Consider the row given by Figure 3(a) from the Example 2.1, we see that the minimum value is − th position of the row, tells us that the row represents vertex 5 in the example. In addition,[ a ij ] = (cid:0) − − − (cid:1) (a) i th row of N M matrix (b) A subgraph corre-sponding to the rowmatrixFigure 3: Row matrix of
N M ( G ) get from Example 2.1 and its graph representation i . The Figure 3(b) shows theconstructed subgraph rooted at vertex 5 by using the corresponding row entries. Further from Figure 3(a),we also get the row sum of N M ( G ) is zero. Proposition 2.6.
For any ≤ i ≤ n , the i th column sum of N M ( G ) is equal to P j ∈ N G ( i ) | N G ( i ) | − | N G ( j ) | ! . Proof.
By Remark 2.4, we have ( MN ) ′ = N M . This implies the column sum of
N M matrix is equal tothe row sum of MN matrix. Therefore, we get n X j =1 η ji = n X j =1 η ′ ij = X j ∈ N G ( i ) | N G ( i ) − N G ( j ) | − X j / ∈ N [ i ] | N G ( i ) ∩ N G ( j ) | − | N G ( i ) | = X j ∈ N G ( i ) | N G ( i ) | − X j ∈ N G ( i ) | N G ( i ) ∩ N G ( j ) | − X j / ∈ N [ i ] | N G ( i ) ∩ N G ( j ) | − | N G ( i ) | = X j ∈ N G ( i ) | N G ( i ) | − | N G ( j ) | ! (by Proposition 2 .
3. 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 forwhich N M ( G ) will be symmetric. Proposition 3.1.
The
N M ( G ) is symmetric if and only if the graph G is either regular or contains regularcomponents.Proof. Let G be a graph with w ( G ) components say G , G , ..., G w such that each G z is regular with degree r z , ≤ z ≤ w ( G ) By the definition of N M ( G ) when i is not adjacent to j then η ij = η ji and when i isadjacent to j then η ij = | N G ( j ) | − | N G ( i ) ∩ N G ( j ) | = r z − | N G ( i ) ∩ N G ( j ) | (4) η ji = | N G ( i ) | − | N G ( i ) ∩ N G ( j ) | = r z − | N G ( i ) ∩ N G ( j ) | (5)From (4) and (5) we have η ij = η ji . Therefore the N M ( G ) is symmetric when the graph G has regularcomponents.Conversely, let the N M ( G ) be symmetric. We know that, N M ( G ) can be written as AD − A . Sincesum of symmetric matrices is symmetric and AD = N M + A , we must have AD to be symmetric. Butfrom 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 , µ ), if G is a k -regular graph on n vertices in which every pair of adjacent vertices has µ common neighbours and everypair of non-adjacent vertices has µ common neighbours. Proposition 3.2.
If a graph G is strongly regular then the entries of N M ( G ) contains either two or threedistinct values. roof. By the definition of
N M ( G ) it immediate follows that for a strongly regular graph G , η ij ( G ) = − k, if i = jk − µ , if ( i, j ) ∈ E ( G ) − µ , if ( i, j ) / ∈ E ( G )where µ = | N G ( i ) ∩ N G ( j ) | , for ( i, j ) ∈ E ( G ) and µ = | N G ( i ) ∩ N G ( j ) | , for ( i, j ) / ∈ E ( G ). This implies theentries of N M ( G ) of a strongly regular graph takes values from {− k, k − µ , − µ } or {− k, k − µ } , when k = µ . Remark . Note that the converse of the above proposition need not be true.
Example . Figure 4(a) is the
N M ( G ) containing only three distinct values as entries, namely, {− , , } .Figure 4(b) is the corresponding graph of Figure 4(a). Note that the graph is not a strongly regular graph. N M = [ η ij ] = − − − − − −
20 0 2 − − − − − − − − − − (a) N M matrix corresponding to G (b) A graph GFigure 4: A graph G and its N M matrix.
Proposition 3.3.
If at least one row of
N M ( G ) has no zero entries then the graph G has diameter at most4.Proof. Let i th row of N M ( G ) have no zero entries then by using the two level decomposition definition wehave d G ( i, j ) ≤ , ∀ j ∈ V ( G ) − i. , otherwise, d G ( i, j ) = 3 for some j this implies η ij = 0.Therefore for any j, k ∈ V ( G ) − i we have d G ( i, j ) ≤ d G ( i, k ) ≤ . So, d G ( j, k ) ≤ d G ( j, i ) + d G ( i, k ) ≤ Remark . Note that the converse of the above proposition need not be true. It is well known that thecubic graph on 8 verrtices ( Q ) has diameter 3 but every row of N M ( Q ) contains exactly one zero. Proposition 3.4.
The
N M ( G ) has no zero entries if and only if the graph G has diameter at most 2.Proof. N M ( G ) has no zero entries ⇐⇒ for every i , i th row of N M ( G ) has no zero entries ⇐⇒ ∀ i, j, i = j, d G ( i, j ) ≤ , ⇐⇒ diameter ( G ) ≤ Proposition 3.5.
The graph G is triangle-free if and only if η ij = | η jj | ∀ ( i, j ) ∈ E ( G ) .Proof. Graph G is triangle-free ⇐⇒ N G ( i ) ∩ N G ( j ) = ∅ for ( i, j ) ∈ E ( G ). By the definition of N M ( G ) if i is adjacent to j then η ij = | N G ( j ) − N G ( i ) | = | N G ( j ) | − | N G ( i ) ∩ N G ( j ) | = | N G ( j ) | . Now in the i th row η ij = | N G ( j ) | = | η jj | . Proposition 3.6.
Given a graph G , the number of triangles in G is given by X i X j ∈ N G ( i ) | η jj | − η ij ! . roof. 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. ∴ Number of triangle containing the vertex i is given by N T ( i ) = 12 P j ∈ N G ( i ) | N G ( i ) ∩ N G ( j ) | , since atriangle < i, j, x, i > will be counted twice, one for each j, x ∈ N G ( i ). Hence,Total number of triangles in the graph = 13 X i N T ( i )= 16 X i X j ∈ N G ( i ) | N G ( i ) ∩ N G ( j ) | = 16 X i X j ∈ N G ( i ) | N G ( j ) | − | N G ( j ) − N G ( i ) | ! = 16 X i X j ∈ N G ( i ) | η jj | − η ij ! Hence the claim.
Remark . It is well known that number of triangle in a graph is equal to 16
T race ( A ) or 16 n P i =1 λ i where A is the adjacency matrix of the graph and λ i , ≤ i ≤ n is the eigenvalue of A . Note that if we want tocount a triangle using the N M ( G ) the computational time involved is very less when compared to compute16 T race ( A ) or 16 n P i =1 λ i Proposition 3.7.
Given a graph G , the number of 4 cycles(including induced and non-induced) is equal to n P i =1 P j ∈ N G ( i ) (cid:0) | η jj |− η ij (cid:1) + P j / ∈ N G ( i ) (cid:0) | η ij | (cid:1)! Proof.
Given a graph G , the number of 4-cycles containing the vertex i is given by P j =1 ,j = i (cid:0) | N G ( i ) ∩ N G ( j ) | (cid:1) .Hence the total number of 4-cycles (both induced and not induced) can be given by,14 n X i =1 X j =1 ,j = i (cid:18) | N G ( i ) ∩ N G ( j ) | (cid:19) = 14 n X i =1 X j ∈ N G ( i ) (cid:18) | N G ( i ) ∩ N G ( j ) | (cid:19) + X j / ∈ N G ( i ) (cid:18) | N G ( i ) ∩ N G ( j ) | (cid:19)! = 14 n X i =1 X j ∈ N G ( i ) (cid:18) | η jj | − η ij (cid:19) + X j / ∈ N G ( i ) (cid:18) | η ij | (cid:19)! Remark . Note that in the above proof, 14 n P i =1 P j / ∈ N G ( i ) (cid:0) | η ij | (cid:1) gives a count of the total number of induced C plus half the number of K − { e } . Similarly, 14 n P i =1 P j ∈ N G ( i ) (cid:0) | η jj |− η ij (cid:1) gives the total number of K along with half the number of K − { e } inthe graph. Proposition 3.8.
A graph G is C -free if and only if η ij ≥ − , ∀ ( i, j ) / ∈ E ( G ) . roof. By the definition of
N M ( G ), we can conclude η ij ≥ − ⇐⇒ | N G ( i ) ∩ N G ( j ) | ≤ , ( i, j ) / ∈ E ( G ) ⇐⇒ G has no induced C Recall that the girth of a graph is the length of a shortest cycle contained in the graph.
Proposition 3.9.
A graph G has girth at least 5 if and only if η ij = | η jj | , ∀ ( i, j ) ∈ E ( G ) and η ij ≥− , ∀ ( i, j ) / ∈ E ( G ) Proof.
By Proposition 3.5 we get, η ij = | η jj | , ∀ ( i, j ) ∈ E ( G ) ⇐⇒ G is Triangle free . and by Proposition 3.8 we get, η ij ≥ − ⇐⇒ | N G ( i ) ∩ N G ( j ) | ≤ , ( i, j ) / ∈ E ( G ) ⇐⇒ G has no induced C Therefore we can conclude G has girth atleast 5.
4. Conclusion
In this paper, we have introduced a new graph matrix (
N M ( G )) that can be associated with a graph toreveal more information when compared to the adjacency matrix. We have also systematically demonstratedthe equivalence of the N M ( G ) and the product of two other existing graph matrices, namely adjacency andLaplacian matrices. Further, we have endorsed its relationship with the concept of level decomposition ofgraph.Further, we have also substantiated the usefulness of the N M ( G ) by identifying numerous propertiesthat can be revealed with the aid of this matrix. In this process, we have found that many simple propertiessuch as counting the number of triangles in a graph can be done in no-time.As an extension of this current work, in a sequel, our subsequent research article comprises of the studyof the N M spectrum. We are also analyzing the algorithmic properties of this matrix and other interestinggraph properties that can be revealed.
In our first attempt to analyze a new graph matrix, we have only studied its correctness and very fewgraph and matrix properties in this paper. This graph matrix seems to be quite promising and can beapplicable in studying problems relating to domination in graphs and graph isomorphism problem. We havealready initiated our study in this direction.
Acknowledgements
The authors would like to acknowledge and thank DST-SERB Young Scientist Scheme, India [GrantNo. SB/FTP/MS-050/2013] for their support to carry out this research at SRM Research Institute, SRMUniversity. Mr. K. Sivakumar would also like to thank SRM Research Institute for their support during thepreparation of this manuscript. 9 eferences [1] R.B. Bapat, Graphs and matrices,
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