In this note we define two generalizations of the line graph and obtain some
results. Also, we mark some open problems

Permi, S.

Prashanth, B.

Reddy, P. Siva Kota Kavita

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A Note On Line Graphs

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International J.Math. Combin. Vol.1 (2011), 119-122A Note On Line GraphsP. Siva Kota Reddy, Kavita. S. Permi and B. PrashanthDepartment of Mathematics, Acharya Institute of TechnologySoladevanahalli, Bangalore-560 090, IndiaE-mail: pskreddy@acharya.ac.inAbstract: In this note we define two generalizations of the line graph and obtain someresults. Also, we mark some open problems.Key Words: Smarandachely (k, r) line graph, (k, r)-graphs, line graphsAMS(2010): 05C75, 05C45For standard terminology and notion in graph theory we refer the reader to Harary [2]; thenon-standard will be given in this paper as and when required. We treat only finite simplegraphs without self loops and isolates.The line graph L(G) of a graph G is defined to have as its vertices the edges of G, with twobeing adjacent if the corresponding edges share a vertex in G. Line graphs have a rich history.The name line graph was first used by Harary and Norman [3] in 1960. But line graphs werethe subject of investigation as far back as 1932 in Whitney s paper [7], where he studied edgeisomorphism and showed that for connected graphs, edge-isomorphism implies isomorphismexcept for K3 and K1,3. The first characterization (partition into complete subgraphs) wasgiven by Krausz [5]. Instead, we refer the interested reader to a somewhat older but still anexcellent survey on line graphs and line digraphs by Hemminger and Beineke [4]. An excellentbook by Prisner [6] describes many interesting generalizations of line graphs. In this note wegeneralize the line graph L(G) of G as follows:Let G = (V,E) be a graph of order p ≥ 3, k and r be integers with 1 ≤ r < k ≤ p. LetU = {S1, S2, ..., Sn} be the set of all distinct connected acyclic subgraphs of G of order k andU ′ = {T1, T2, ..., Tm} be the set of all distinct connected subgraphs of G with size k.The vertex (k, r)-graph Lv(k,r)(G), where 1 ≤ r < k ≤ p is the graph has the vertex setU where two vertices Si and Sj , i 6= j are adjacent if, and only if, Si ∩ Sj has a connectedsubgraph of order r.The edge (k, r)-graph Le(k,r)(G), where 0 ≤ r < k ≤ q is the graph has the vertex set U ′where two vertices Ti and Tj , i 6= j are adjacent if, and only if, Ti∩Tj has a connected subgraphof size r.The Smarandachely (k, r) line graph LS(k,r)(G) of a graph G is such a graph with vertexset U ′ and two vertices Ti and Tj , i 6= j are adjacent if and only if, Ti ∩ Tj has a connectedsubgraph with order or size r. Clearly, Lv(k,r)(G) ≤ LS(k,r)(G) and Le(k,r)(G) ≤ LS(k,r)(G). In1Received December 22, 2010. Accepted March 3, 2011.120 P. Siva Kota Reddy, Kavita. S. Permi and B. PrashanthFigure 1, we depicted L(G), Lv(k,r)(G) and Le(k,r)(G) for the graph G.Figure.1One can easily verify that: L3(K1,3) = K3, L3(Cn) = Cn, for n ≥ 3, L3(Pn) = Pn−1 andL3(K4) = L(K4 − e) = K4.For any positive integer k, the kth iterated line graph L(G) of G is defined as follows:L0(G) = G, Lk(G) = L(Lk−1(G)).A graph G is a (3, 2)-graph if there exists a graph H such that L(3,2)(H) ∼= G. First weprove the following result:Proposition 1 For any graph G, L(3,2) ∼= L2(G).Proof First we show that L(3,2)(G) and L2(G) have the same number of vertices. Let Sbe a vertex in L(3,2)(G). Then S corresponds to a subgraph of order 3 in G. Say, S consistsof two adjacent edges ab and bc. Then corresponding to S we have an edge in L(G) with endvertices ab and bc, and corresponding this edge, we have a vertex say abc in L2(G). Similarly,we can show that very vertex in L2(G) corresponds to a connected subgraph of order 3. Thisproves that L(3,2)(G) and L2(G) have the same number of vertices. Now, let S1 and S2 be twoadjacent edges in L(3,2)(G). Then S1 and S2 correspond to two connected subgraphs of order3 each, having a common edge. These in turn will give two adjacent edges, say e(S1) and e(S2)in L(G) and this will give an edge in L2(G) with end vertices e(S1) and e(S2). This proves theresult. In general one can establish the following result.Proposition 2 Let G be a graph of order p and 2 ≥ r < k ≤ p. If △(G) ≤ 2 then L(k,k−1)(G) ∼=Lk−1(G).Note that this is true only when △(G) ≤ 2. For example, we find Ln−1(n,n−1)(K1,n) = Kn =L(K1,n).Proposition 3 The star K1,3 is not a 3-line graph.A Note On Line Graphs 121Proof Let K1,3 be a 3-line graph. Then there exists a graph G such that L(K1,3) = G.Since K1,3 has four vertices, G should have exactly four connected subgraphs each of orderthree. All connected graphs having exactly four induced subgraphs are as follows: i) C4, ii) K4,iii) K4 − e and iv) P6. None of these graphs have K1,3 as its 3-line graph. Clearly, any graph having K1,3 as an induced subgraph is not a 3-line graph. Hence, wehaveCorollary 4 K1,n, n ≥ 3 is not a 3-line graph.It is not true in general that the line graph L(G) of a graph G is a subgraph of L3(G). Forexample, L3(K1,4) does not contain K4, the line graph of K1,4 as a subgraph.Problem 5 Characterize 3-line graphs.A graph G is a self 3-line graph, if it is isomorphic to its 3-line graph.Problem 6 Characterize self 3-line graphs.Proposition 7 Let L3(G) be the 3-line graph of a graph G of order p ≥ 3. The degree of avertex s in L3(G) is denoted by degs and is defined as follows:Let S be the subgraph of G corresponding to the vertex s in L3(G). For an edge x = uv inS, let d(x) = (degGu+ degGv)− (degSu+ degSv), where degGu and degSu are the degrees of uin G and S respectively. Then d(s) =∑x∈Sd(x).Proof Consider an edge uv in S. Suppose y = uz is an edge of G at u which is not in S.Then y belongs to a connected subgraph S1 of cardinality three containing the edge uv whichis distinct form S. Since S and S1 have common edge, ss1 is an edge in L3(G), where s1 is thevertex in L3(G) corresponding to the subgraph S1 in G. Similarly, for any edge y1 = vz1 at vin G which is not in S, we have an edge ss2 in L3(G). This implies that corresponding to theedge x = uv in S, we have (degGu− degSu) + (degGv − degSv) edges in L3(G), and hence theresult follows. AcknowledgementThe authors are grateful to Sri. B. Premnath Reddy, Chairman, Acharya Institutes, for hisconstant support and encouragement for research and development.References[1] F.Barahona, M.Grotschel, M.Junger, and G.Reinelt, An application of combinatorial opti-mization to statistical physics and circuit layout design, Operations Research, 36 (3) (1988),493-513.[2] F. Harary, Graph Theory, Addison-Wesley Publishing Co., 1969.122 P. Siva Kota Reddy, Kavita. S. Permi and B. Prashanth[3] F.Harary and R. Z. Norman, Some properties of line digraphs, Rend. Circ. Mat. Palermo,(2) 9 (1960), 161-168.[4] R. L.Hemminger and L. W. Beineke, Line graphs and line digraphs, Selected Topics inGraph Theory (W. B. Lowell and R. J. Wilson, eds.), Academic Press, New York, 1978,pp. 271-305.[5] J.Krausz, De´monstration nouvelle d u ne the´ore`me de Whitney sur les re´seaux, Mat. Fiz.Lapok, 50 (1943), 75-85 (Hungarian).[6] E. Prisner, Graph Dynamics, Pitman Research Notes in Mathematics Series, vol. 338,Longman, Harlow, 1995.[7] H. Whitney, Congruent graphs and the connectivity of graphs, Amer. J. Math., 54 (1932),150-168.

Congruent graphs and the connectivity of graphs,

De´monstration nouvelle d u ne the´ore`me de Whitney sur les re´seaux,

Graph Dynamics, Pitman Research Notes

Graph Theory,

Line graphs and line digraphs, Selected Topics in Graph Theory

M.Junger, and G.Reinelt, An application of combinatorial optimization to statistical physics and circuit layout design,

Some properties of line digraphs,

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A Note On Line Graphs

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