Common Neighborhood Product of Graphs

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Volume 114 No. 6 2017, 203-208 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu ijpam.

1 Volume 114 No , ISSN: (printed version); ISSN: (on-line version) url: ijpam.eu Common Neighborhood Product of Graphs A Babu 1 and J Baskar Babujee 2 1,2 Department of mathematics, Anna University, MIT Campus, Chennai , India. 1 subashbabu18@gmail.com 2 baskarjee@annauniv.edu February 28, 2017 Abstract In this paper we introduce the Common Neighborhood Product of any two arbitrary graphs and analyze characterization of this graph product. AMS Subject Classification: 05C76 Key Words and Phrases:Common Neighborhood Graphs, Common Neighborhood Graph Products. 1 Introduction The concepts of visualising graph product was stimulated from a biological model proposed by Wagner and Stadler [3] that provided a impression concerning the topological theory of the relationships between genotypes and phenotypes. A visualization for graph products is needed that can effectively communicate the quality of results by emphasizing the regularity of graph structure through regularity of layout. Other areas where graph products play vital role can be found in design and analysis of networks, computational engineering. For example the formation of finite element models or construction of localized self-equilibrating systems in computational engineering, see [6], [7], [8]. Typical tasks in scientic computing, like solving discretized partial differential equations,need 203 1

2 computational meshes. Many Graph products have been studied like Cartesian product, tensor product, Strong product of graphs, Rooted product of graphs, etc, and also studied their characteristic like connectivity,diameter, minimum degree, maximum degree, Chromatic number, energy etc,. Let G be a simple graph with vertex set V (G) and edge set E(G). The common neighborhood graph (congraph) of G denoted by con(g) is the graph with V (con(g)) = V (G), in which two vertices of con(g) adjacent if they have common neighbor in G. A. Alwadi et al introduced the common neighborhood graphs from the motivation on theory of graph energy [1],[2]. Some basic properties of congraphs have been studied in [4] are given below. Theorem 1. The common neighborhood graphs con(g) is connected if and only if the parent graph G is connected and nonbipartite. Corollary 2. if G is a connected bipartite graph, then con(g) has exactly two components. Theorem 3. If G has degree sequence d 1, d 2,...d n and m is the number of edges of con(g), then m n ( di ) i=1 2 and equality holds if and only if G is quadrangle free. A strongly regular graph with parameters (n, k, s, t) is a k- regular graph with n vertices, such that any two adjacent vertices have s common neighbors and any two non-adjacent vertices have t common neighbors. The congraph of any strongly regular graph with s 0 is the complete graph K n Theorem 4. The congraph of all strongly regular graphs with parameter (n, k, 0, t) except of C 5 are hyperenergetic In this paper we define a new graph product called common neighborhood product and study some basic properties of the common neighborhood product of graphs

3 2 Main Results All graphs we consider are simple and finite. Readers can follow basic graph theoretical terminology and notation from [5] which is not defined here. Definition 5. Let G and H be two graphs. The Common neighborhood product of graphs denoted by GΘH has vertex set V (GΘH) = V (G) V (H) and two vertices (a,b) (c,d) are adjacent in GΘH if a adjacent to c in G and vertices b and d have common neighbor in H or b adjacent to d in H and vertices a and c have common neighbor in G. v 3 u3 u2 v 1 v 2 G H u1 ( v1, u1) ( v1, u2) ( v1, u3) ( v2, u1) ( v2, u2) ( v2, u3) ( v3, u1) ( v3, u2) ( v3, u3) Figure 1: Common Neighbourhood Product of G and H Clearly GΘH and HΘG are naturally isomorphic, but it is not commutative as an operation on labeled graphs. GΘH is also associative as the graphs (F ΘG)ΘH and F Θ(GΘH) are isomorphic to each other and GΘH is n-partite graph. Theorem 6. If m and n are number of edges of G and H respectively, then number of edges of GΘH = 2(m E(con(H) )

4 n E(con(G) ). Proof. Let G and H be two graphs with m and n edges respectively. Consider {v 1, v 2 } V (G) and {u 1, u 2 } V (H). Then each (v 1, u 1 )(v 2, u 2 ) E(GΘH) if v 1 adjacent to v 2 in G and the vertices u 1 and u 2 have common neighbor in H. Each vertex v i adjacent to v j in G and each vertex pair u k and u l having common neighbor in H will contribute m E(con(H) edges to GΘH. Then v 2 also adjacent to v 1 in G therefore (v 2, u 1 )(v 1, u 2 ) E(GΘH). Hence m edges of G contribute 2m E(conH) to GΘH. Similarly n edges of H and edges of con(g) contribute 2n E(conG) edges to GΘH. Which conclude the proof. Corollary 7. multiple of four. Number of odd degree vertices in GΘH is always Theorem 8. If m and n are number of vertices of the G and H respectively, then (GΘH) mn (m + n) + 1. Proof. Consider the graph G and H which has m and n vertices respectively, therefore V (GΘH) = mn. Since G and H are simple and no two vertices of {(v k, u i ), (v k, u j )}; 1 k m and 1 i, j n of GΘH are adjacent. Therefore m vertices of GΘH are never adjacent in GΘH even u i and u j have common neighbor in H. similarly no two vertices {(v i, u k ), (v j, u k )}; 1 i, j m and 1 k n of GΘH is adjacent. Therefore n vertices of GΘH are never adjacent in GΘH. Clearly there are m+n 1 vertices never adjacent to each other in GΘH. Hence (GΘH) mn (m + n) + 1. Theorem 9. If GΘH is connected then δ(gθh) 2 Proof. Since GΘH is connected therefore G and H must be connected and number of vertices of G and H are greater than or equal to three therefore there exist atleast one pair of vertices have common neighbour in G and H respectively. Consider the vertex (u i, v j ) V (GΘH). Each u i is adjacent to some vertex of G and H has atleast one pair of vertices has common neighbor therefore (u i, v j ) must adjacent to atleast one of the vertices of GΘH. Similarly each v j is adjacent to some vertex of H and G is having atleast one common neighbour therefore (u 1, v 1 ) is adjacent to one of vertices of (u i, v j ). Hence every vertex of GΘH has atleast 206 4

5 two connected. Theorem 10. If GΘH is connected then the dia(gθh) 2. Proof. Since GΘH is connected therefore every vertex of GΘH is connected by a path. Let (u i, v j ) V (G) and no two vertices of (u i, v j ) and (u i, v k ) are adjacent in GΘH. Similarly no two vertices of (u i, v j ) and (u k, v j ) are adjacent ingθh. Therefore there exist a path connecting these vertices whose length is greater than one. Theorem 11. {dia(g), dia(h)}. If GΘH is connected then dia(gθh) max Proof. Suppose d G (a, c) = m and d H (b, d) = n. Let P 1 = a = x 0, x 1...x m = c and P 2 = b = y 0, y 1...y m = d therefore there exist the following paths between a to c with length m that is (x 0, y 0 ), (x 1, y 2 ), (x 2, y 0 ),..., (x m, y 0 ), (x m, y 2 ) (x 0, y 1 ), (x 1, y 3 ), (x 2, y 1 ),..., (x m, y 1 ), (x m, y 3 ),..., (x 0, y n 1 ), (x 1, y n ), (x 2, y n 1 ),..., (x m, y n 1 ), (x m, y n ) Similarly, there exist the following paths between b to d with length n that is (x 0, y 1 ), (x 2, y 2 ), (x 0, y 3 ),..., (x 0, y n 1 ), (x 2, y n ) (x 1, y 0 ), (x 3, y 1 ), (x 1, y 2 ),..., (x 1, y n 1 ), (x 3, y n ),..., (x n 1, y 0 ), (x n, y 1 ), (x n 1, y 2 ),..., (x n 1, y n 1 ), (x n, y n ) therefore distance between any two vertices in GΘH is not more than m or n. Hence dia(gθh) max {dia(g), dia(h)} Lemma 12. Suppose G and H are connected graphs with at least three vertices then GΘH has atmost two components. Theorem 13. If GΘH is connected then δ(gθh) δ(g) + δ(h). Proof. Let δ(g) = m and δ(h) = n and each vertices of G has degree atleast m. Let (u i, v j ) GΘH and u i be a vertex of G which has degree atleast m therefore u i must be adjacent to m vertices of G. Since GΘH is connected, there exist atleast one pair of vertices has common neighbour in H. Therefore the vertex (u i, v j ) must be connected to m vertices. Similarly, each vertices of H has degree atleast n and atleast one pair of vertices has common neighbor. Therefore the vertex (u i, v j ) must adjacent to some n 207 5

6 vertices. Hence each vertices of degree atleast m + n. Theorem 14. κ(g) + κ(h). For connected graph G and H κ(gθh) Proof. Let δ(g) = m and δ(g) = n. The whitney inequality states that κ(g) λ δ(g) previous lemma shows that δ(gθh) δ(g) + δ(h). Hence we can conclude that κ(gθh) κ(g) + κ(h) References [1] A. Alwardi, N.D. Sonar, I.Gutman, N.M. M de Avreu Complete common neighborhood graphs, Bull Acad.Serbe Sci. Art(Cl. Sci. Math)(2011), [2] A. Alwardi, B. Arsic, I. Gutman, N. D. Sonar, The common neighborhood graph and its energy, Iran. J. Math. Inf. (2012) 7(2) 1-8. [3] G. D. Battista, P. Eades, R. Tamassia, and I. Tollis. Graph Drawing. Prentice Hall, [4] A. S. Boniffacia, R. R. Rosa, I. Gutman, N. M. M. De Abreu, Complete common neighborhood graphs, proceedings of Congreso Latino-Ibroamericano de Investigation Operativa and Simposia Brasilerio de pesquisa operacional, (2012) [5] J. A. Bondy and U. S. R. Murthy, Graph Theory with Application, London and Basingstoke, macmillan Press Ltd [6] A. Kaveh and H. Rahami. An efcient method for decomposition of regular structures using graph products. Intern. J. for Numer. Methods in Engineering,(2004) 61(11) [7] A. Kaveh and K. Koohestani. Graph products for conguration processing of space structures. Comput. Struct.,(2008) 86(11-12) [8] A. Kaveh and R. Mirzaie. Minimal cycle basis of graph products for the force method of frame analysis. Communications in Numerical Methods in Engineering,(2008) 24(8)

CLASSES OF VERY STRONGLY PERFECT GRAPHS. Ganesh R. Gandal 1, R. Mary Jeya Jothi 2. 1 Department of Mathematics. Sathyabama University Chennai, INDIA

CLASSES OF VERY STRONGLY PERFECT GRAPHS. Ganesh R. Gandal 1, R. Mary Jeya Jothi 2. 1 Department of Mathematics. Sathyabama University Chennai, INDIA Inter national Journal of Pure and Applied Mathematics Volume 113 No. 10 2017, 334 342 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu ijpam.eu Abstract: CLASSES