International Journal for Research in Engineering Application & Management (IJREAM) ISSN : 2454-9150 Vol-04, Issue-09, Dec 2018 Adjacent Line Graphs of Simple Graphs Dr. B.Vasudevan, Assistant Professor of Mathematics, Thiru.Vi.Ka.Govt.Arts College, Tiruvarur, [email protected] K.Vimala, Assistant Professor of Mathematics, STET Womens’ College, Mannargudi, Tamilnadu, [email protected] Abstract The concept of line graph was discovered by many authors independently and they gave it a different names, like derived graph[4], interchange graph[9], edge graph and edge-to-vertex dual. But the name line graphs comes from a paper by Harary &Norman(1960)[6]. In [1] Bagga, Beineke and Varma introduced the concept of super line graphs. In this paper, we introduce a new graph, called adjacent line graph, associated with a given simple graph G. Also the adjacent line graphs of Path Cycle and Star are discussed. Keywords — Graphs, simple graphs, regular graphs, complete graphs, line graphs. AMS(2010): 05C75, 05C45 I. INTRODUCTION EXAMPLE 1.2 Let G (V, E) be a graph with V = { } and E = { }. The edge graph or line graph of G is denoted by L(G) has the vertex set E and two vertices and are adjacent in L(G) if and only if the corresponding edges and of G are adjacent in G. That is, in the line graph L(G) each vertex corresponds to a pair of adjacent vertices in G. In this paper we introduce a new graph, called EXAMPLE 1.3 the adjacent line graph, associated with a graph G, in which each vertex corresponds to a pair of adjacent edges in G. II. ADJACENT LINE GRAPHS DEFINITION 1.1 Let G = (V, E) be a simple graph with atleast one pair of adjacent edges. The adjacent line graph of G, denoted by AL(G), is a graph with the vertex set = { XAMPLE / are adjacent in G } and two vertices , E 1.4 are adjacent in AL(G), if and only if, either the edges & or & or or & are adjacent in G. EXAMPLE 1.1 EXAMPLE 1.5 389 | IJREAMV04I0945074 DOI : 10.18231/2454-9150.2018.1207 © 2018, IJREAM All Rights Reserved. International Journal for Research in Engineering Application & Management (IJREAM) ISSN : 2454-9150 Vol-04, Issue-09, Dec 2018 REMARK 1.1 We continue this process until there is no edge is left in G. Then we have a path between every pair of vertices in 1. If are vertices of G, then the AL(G). Hence AL(G) is connected. number of vertices in AL(G) is equal to ∑ ( ), Conversely assume that AL(G) is connected. We have to where is the degree of the vertex . prove that G is connected. Suppose G is not connected. Without loss of generality, we assume that G has two 2. The number of vertices in AL( AL( , components and . Then the adjacent line graph of G AL( and AL( are respectively has two components and hence AL(G) is disconnected. and . This is a contradiction to our assumption that AL(G) is connected. Hence G must be connected. 3. The number of vertices in AL( is . THEOREM 1.2 REMARK 1.2 The adjacent line graph of the cycle is 4- regular. There are graphs for which the adjacent line graphs are same. The following examples illustrate this. PROOF EXAMPLE 1.6 Let = . Then the vertices of AL( are denoted by , , ,…….., . Consider the vertex in AL( . It is adjacent to the four vertices and . So the degree of the vertex is 4. Similarly the vertex is adjacent to , and and the vertex is adjacent to , and . Thus all EXAMPLE 1.7 the vertices of AL( are of degree 4. Hence AL( is 4-regular. REMARK 1.3 The adjacent line graph of is and the adjacent line graph of is . THEOREM 1.3 The adjacent line graph of the star is the complete graph . i.e., AL( = . THEOREM 1.1 A graph G is connected if and only if AL(G) is PROOF connected. The number of vertices in the adjacent line graph of the star is , since the degree of one vertex in is and PROOF all the other vertices are pendant vertices. Since every edge The theorem is clearly true for a graph G with two edges, in is adjacent to every other edges, we have the degree since these two edges make a vertex in AL(G). of each vertex in AL( is . Hence the adjacent line graph of the star is the complete graph . Let us assume that G is a connected graph with m ( 3) edges. Since G is connected, there exists a path between every pair of vertices. Thus , by definition of adjacent line REMARK 1.4 graph, for every pair of adjacent edges in G we have a The number of edges in AL( is . vertex in AL(G). Let be an arbitrary vertex in AL(G). That is, and are adjacent edges in G. Here, atleast one of them must have an adjacent edge in G, since G is connected. This leads to another vertex or in AL(G). 390 | IJREAMV04I0945074 DOI : 10.18231/2454-9150.2018.1207 © 2018, IJREAM All Rights Reserved. International Journal for Research in Engineering Application & Management (IJREAM) ISSN : 2454-9150 Vol-04, Issue-09, Dec 2018 THEOREM 1.4 [7] F. Harary, Graph Theory, Addison-Wesley Publishing Co., 1969. If G is the graph , then AL(G) contains exactly two vertices of degree 2, two vertices of degree 3 and the [8] R. L. Hemminger and L. W. Beineke, Line Graphs and remaining vertices are of degree 4. line digraphs, Selected Topics in Graph Theory (W. B. Lowell and R. J. Wilson, eds.), Academic Press, PROOF New York, 1978, pp. 271-305 Let G be the path = [9] Van Rooij A. C. M,; Wilf, H.S (1965), “The . Then the interchange graph of a finite graph”, Acta Mathematica vertices of AL( are denoted by , , …….., Hungarica 16(3-4); 263-269. Consider the vertex . It is adjacent to the four vertices and . So the degree of the vertex is 4. But the vertices and are adjacent to two vertices & and & respectively. Hence they are of degree 2. The vertices and are adjacent to three vertices , & and & respectively. Hence they are of degree 3. Since the number of vertices in AL( is , the number of vertices of degree 4 in AL( is . COROLLARY 1.1 AL( contains an Euler line. III. CONCLUSION In this paper, the concept of adjacent line graph, associated with a simple graph G, is introduced. We have proved that the adjacent line graph of cycle is 4-regular and the adjacent line graph of star is a complete graph. The adjacent line graph of Path contains an Euler line. IV. REFERENCES [1] J.S Bagga, L.W. Beineke and Varma, Super line graphs and their properties, Combinatorics, Graph Theory, Algorithm and Applications(Beijing, 1993), World Scientific Publishing, New Jersey, 1994, pp. 1-6. [2] J.S Bagga, L.W. Beineke and Varma, The Super line graph , Discrete Math. 206 (1999), no. 1-3, 51-61 [3] R. Balakrishnan and K. Ranganathan, A Textbook of Graph Theory, Universitext, Springer-Verlag, New York, 2000. [4] L. W. Beineke, Charaterizations of derived graphs, J. Combinatorial Theory 9 (1970), 129-135. [5] J.A. Bondy and U.S.R.Murthy, Grph Theory with Applictions, Macmillan, London and Elseveir, New Yark (1976). [6] Harary F & Norman, R Z(1960), “Some properties of line digraphs” Rendiconti del Circolo Mathematico di Palermo9(2); 161-169 391 | IJREAMV04I0945074 DOI : 10.18231/2454-9150.2018.1207 © 2018, IJREAM All Rights Reserved. .