IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 13, Issue 1 Ver. IV (Jan. - Feb. 2017), PP 01-07 www.iosrjournals.org DOI: 10.9790/5728-1301040107 www.iosrjournals.org 1 | Page Connected and Distance in G ⊗ 2 H U. P. Acharya 1 , H. S. Mehta 2 1 (Department of Appl. Sci. and Humanities, A. D. Patel Institute of Technology, New V. V. Nagar, India) 2 (Department of Mathematics, S. P. University, V. V. Nagar-388120, India) Abstract : The tensor product G H  of two graphs G and H is well-known graph product and studied in detail in the literature. This concept has been generalized by introducing 2-tensor product 2 G H  and it has been discussed for special graphs like n P and n C [5]. In this paper, we discuss 2 G H  , where G and H are connected graphs. Mainly, we discuss connectedness of 2 G H  and obtained distance between two vertices in it. Keywords: Bipartite graph, Connected graph, Non-bipartite graph, 2-tensor product of graphs. I. Introduction The tensor product G H  of two graphs G and H is very well-known and studied in detail ([1], [2], [3], [4]). This concept has been extended by introducing 2  tensor product 2 G H  of G and H and studied for special graphs [5]. In this paper, we discuss connectedness of 2 G H  for any connected graphs G and H . We also obtained the results for the distance between two vertices in 2 G H  . If =( ( ), ( )) G VG EG is finite, simple and connected graph, then (, ) G d uu is the length of the shortest path between u and u in G . For a graph G , a maximal connected subgraph is a component of G . For the basic terminology, concepts and results of graph theory, we refer to ([1], [6], [7]). We recall the definition of 2  tensor product of graphs. Definition 1.1 [5] Let G and H be two connected graphs. The 2  tensor product of G and H is the graph with vertex set {( , ): ( ), ( )} uv u VGv VH   and two vertices (,) uv and ( , ) uv   are adjacent in 2  tensor product if (, )=2 G d uu and (, )=2 H d vv . It is denoted by 2 G H  . Note that 2 G H  is a null graph, if the diameter ( )<2 DG or ( )<2 DH . So, throughout this paper we assume that G and H are non-complete graphs. II. Connectedness of 2 G H  this section, first we consider the graphs G and H , both connected and bipartite with 2 ( ) ; ( ) ( ) N w w VG VH      , where 2 ( )={ ( ): (, ) = 2} G N u u VG d uu    In usual tensor product G H  , the following result is known. Proposition 2.1 [4] Let G and H be connected bipartite graphs. Then G H  has two components. Note that in case of 2 G H  , the similar result is not true. We discuss the number of components in 2 G H  with different conditions on G and H . We fix the following notations Let 1 2 ( )= VG U U  and 1 2 ( )= VH V V  with i U and j V , (, = 1, 2) ij are partite sets of G and H respectively. Then, 2 11 12 21 22 ( )= with = ij i j VG H W W W W W U V      Remark 2.2 If (,) uv and ( , ) uv   are from different ij W , then (,) uv and ( , ) uv   can not be adjacent in 2 G H  as (, ) 2 G d uu or (, ) 2 H d vv . So, 2 G H  has at least four components. Suppose (,) uv and ( , ) uv   are in the same ij W . Then (, ) G d uu and (, ) H d vv are even. Proposition 2.3 Let G and H be connected bipartite graphs. If (, ) G d uu and (, ) H d vv are of the same form, 4k or 4 2 k  , ( IN {0}) k   then (,) uv and ( , ) uv   are in the same component of 2 G H  . Proof.Let 1 1 (,)&( , ) uv uv U V    .Suppose, 1 0 1 : = = m Pu u u u u     and 2 0 1 : = = n P v v v v v     are paths between u , u and v , v respectively. Suppose 1 ( )=4 /4 2 lP k k  and 2 ( )=4 /4 2 lP t t  with k t  . First assume that 0 k t   , then there is a path P or P between (,) uv and ( , ) uv   in 2 G H  as follows: