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Distance Product of Graphs Applications and Applied Mathematics: An International Journal (AAM) Volume 13 Issue 1 Article 13 6-2018 Distance Product of Graphs H. S. Mehta Sardar Patel University U. P. Acharya A. D. Patel Institute of Technology Follow this and additional works at: https://digitalcommons.pvamu.edu/aam Part of the Discrete Mathematics and Combinatorics Commons Recommended Citation Mehta, H. S. and Acharya, U. P. (2018). Distance Product of Graphs, Applications and Applied Mathematics: An International Journal (AAM), Vol. 13, Iss. 1, Article 13. Available at: https://digitalcommons.pvamu.edu/aam/vol13/iss1/13 This Article is brought to you for free and open access by Digital Commons @PVAMU. It has been accepted for inclusion in Applications and Applied Mathematics: An International Journal (AAM) by an authorized editor of Digital Commons @PVAMU. For more information, please contact [email protected]. Mehta and Acharya: Distance Product of Graphs Available at Applications and Applied http://pvamu.edu/aam Mathematics: Appl. Appl. Math. An International Journal ISSN: 1932-9466 (AAM) Vol. 13, Issue 1 (June 2018), pp. 190–198 Distance Product of Graphs 1H. S. Mehta and 2U. P. Acharya 1Department of Mathematics Sardar Patel University Vallabh Vidyanagar-388120, India [email protected] 2A. D. Patel Institute of Technology New V. V. Nagar -388121, India [email protected] Received: August 23, 2017; Accepted: March 30, 2018 Abstract In graph theory, different types of product of two graphs have been studied, e.g. Cartesian product, Tensor product, Strong product, etc. Later on, Cartesian product and Tensor product have been generalized by 2−Cartesian product and 2−Tensor product. In this paper, we give one more gen- eralize form, distance product of two graphs. Mainly we discuss the connectedness, bipartiteness and Eulerian property in this product. Keywords: Connected graph; Cartesian product; Tensor product; 12−product of graphs; Bipartite MSC 2010 No.: 05C69, 05C70, 05C76 1. Introduction The Cartesian product and Tensor product of graphs are well-studied by Hammack et al. (2011) and Sampathkumar (1975). Later on 2−Cartesian product and 2−Tensor product are defined and discussed by Acharya et al. (2014, 2015, 2017). These product are defined using the concept of vertices at distance two. The graph with this concept, e.g. the square graph G[2] and the derived graph G0, are also studied in detail. In fact this concept is useful in studying energy of the derived graph which has wide application in chemical graph theory by Ayyaswamy et al. (2010), Hande et al. (2013) and Jog et al. (2012). But the unfortunately 2−Cartesian product and 2−Tensor product 190 Published by Digital Commons @PVAMU, 2018 1 Applications and Applied Mathematics: An International Journal (AAM), Vol. 13 [2018], Iss. 1, Art. 13 AAM: Intern. J., Vol. 13, Issue 1 (June 2018) 191 does not preserve connectedness. So, using two and usual cartesian product and tensor product we define the distance product of graphs in this paper. Mainly, we prove that the new product will give connected. Let G = (V (G);E(G)) be a finite and simple graph with the vertex set V (G) and the edge set E(G). A graph G is connected, if there is a path between every pair of vertices. If G is a connected graph, 0 0 then dG(u; u ) is the length of the shortest path between u and u in G. The diameter of G, denoted 0 0 by D(G); is defined as max fdG(u; u ): u; u 2 V (G)g. The null graph is a graph with empty edge set. Throughout this paper, we fix G and H to be finite, connected and simple graphs. For the basic terminology, concepts and results of graph theory, we refer to Balakrishnan et al. (2012) and Godsil et al. (2011). 2. Distance Product G ⊗12 H In this section we define distance product of two graphs and discuss the connectedness of this new product. Definition 2.1. Let G = (U; E1) and H = (V; E2) be two connected graphs. The distance product or 12−product of G and H, denoted by G ⊗12 H, is the graph with the vertex set U × V and two vertices (u; v) and 0 0 (u ; v ) in U × V are adjacent in G ⊗12 H if 0 0 (i) dG(u; u ) = 1 and dH (v; v ) = 2, or 0 0 (ii) dG(u; u ) = 2 and dH (v; v ) = 1. Note that G ⊗12 H is isomorphic to H ⊗12 G. Also the graph G ⊗12 H is a null graph if D(G) < 2 and D(H) < 2. Example 2.2. (i) Let G = P3 with u1 ! u2 ! u3 and H = P4 with v1 ! v2 ! v3 ! v4. Then P3 ⊗12 P4 is as follows: (u3; v1) (u3; v2) (u3; v3) (u3; v4) H H ¨¨ ¨¨ A HH ¡A H¨H ¡A ¨ ¡ rAA H¡ rA¨ H¡ rA¨ ¡ r A ¨¡ H¨A ¨¡ H¨A ¡ A ¨ H ¨(u ; vH) ¨AA (¡u2; v2¨)AH¡H2 3AH¡H ¨ A¡ ¨ A¡ H A¡ H (u2; v1) HH A HH ¨¨ ¨¨ (u2; v4) ¡HAA ¡H¨A ¡¨A r¡ HA H¨ r¨¡ HA H¨ r¨¡ A r ¨A H ¨ H ¡¨¨ AA ¡¨H¨HA ¡ HHA ¡¨ A¨¡ HA¡ HA (u ; v ) (u ; v ) (u ; v )(u ; v ) r1 1 r1 2 r1 3 r 1 4 https://digitalcommons.pvamu.edu/aam/vol13/iss1/13 2 Mehta and Acharya: Distance Product of Graphs 192 H. S. Mehta and U. P. Acharya (ii) Let G = P3 = H with u1 ! u2 ! u3 and v1 ! v2 ! v3 in G and H respectively. Then the graph P3 ⊗12 P3 is not connected and it is as follows: (u3; v1) (u3; v2) (u3; v3) H ¨ A H ¡A ¨¡ A H ¡ A ¨ ¡ rA H¨ r¨ r AA ¨¡ HHA ¡ A ¨¡ AH¡ (u ; v ) ¨¨A HH 2 1 H AA¡ A¡ ¨ H¡HA(u2; v2)¡¨A¨ (u ; v ) rAH r¨ r 2 3 ¡ A¨AH¨H¡ A ¡ ¨ AA ¡ H A ¡¨¨ A¡ HHA (u1; v1) r(u1 r; v2) r(u1; v3) 2 To obtain the connectedness of G⊗12 H, we fix G and H both are connected graphs with N (w) 6= φ 2 for every w 2 V (G) [ V (H), where N (u) = fv 2 V (G): dG(u; v) = 2g. Theorem 2.3. 2 Let G and H be two graphs with N (w) 6= φ, for every w 2 V (G) [ V (H). Then G ⊗12 H is connected if and only if G and H are connected. Proof: Let G and H be two graphs with vertex sets U and V respectively. Then V (G ⊗12 H) = U × V . First, assume that G and H are connected. 0 0 0 Let (u; v) and (u ; v ) be in U × V . Also, u = u0 ! u1 ! ::: ! um = u is a path in G and 0 v = v0 ! v1 ! ::: ! vn = v is a path in H. 2 0 2 0 0 As N (u) 6= φ in G, 9 x 2 N (u) such that dG(u; x ) = 2. Let u ! x ! x be a path between u and 0 0 2 0 0 0 x in G. Similarly 9 y 2 N (v1) such that dH (v1; y ) = 2. Then P1 :(u; v) ! (x ; v1) ! (x; y ) ! (u; v1) is a path between (u; v) and (u; v1) in G ⊗12 H. By continuing the same process, there is a 0 0 0 0 path between (u; v) and (u; v ) in G ⊗12 H. Similarly, there is a path between (u; v ) and (u ; v ) in 0 0 G ⊗12 H. Using these paths, we get a path between (u; v) and (u ; v ) in G ⊗12 H. Hence, G ⊗12 H is a connected graph. 0 0 Conversely, suppose that G ⊗12 H is connected. Let u and u be in V (G) with u 6= u . 0 0 0 0 Let P :(u; v) = (u0; v0) ! (u1; v1) ! ::: ! (un; vn) = (u ; v ) be a path between (u; v) and (u ; v ) 0 in G ⊗12 H for v; v 2 V (H). Now, from path P , we have (u0; v0) ! (u1; v1) which gives that dG(u0; u1) = 1 or 2 in G. Similarly, for any two adjacent vertices (ui; vi) ! (ui+1; vi+1) on path P 0 gives that dG(ui; ui+1) = 1 or 2 in G. So, there is a walk between u and u in G. Hence, there is a path between u and u0 in G. Therefore G is a connected graph. By similar arguments, H is also a connected graph. Next, we consider the case in which only one of the graph has property N 2(u) 6= φ, for every u. Published by Digital Commons @PVAMU, 2018 3 Applications and Applied Mathematics: An International Journal (AAM), Vol. 13 [2018], Iss. 1, Art. 13 AAM: Intern. J., Vol. 13, Issue 1 (June 2018) 193 Theorem 2.4. 2 Let G be a non-complete graph and H be a graph with N (b) 6= φ; 8 b 2 V (H). Then G ⊗12 H is connected if and only if G and H are connected. Proof: Assume that G and H are connected graphs. We continue the notations of Theorem 2.3. Let (u; v) and (u0; v0) be in U × V . If N 2(u) 6= φ in G, then we get the path between (u; v) and (u0; v0) as in Theorem 2.3. 2 0 0 Suppose N (u) = φ. Then dG(u; u ) = 1 for every u 2 V (G). In particular dG(u; u1) = 1.
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