РЕБЕРНЫЙ $C_k$-ГРАФ ГРАФА Текст научной статьи по специальности «Математика»

Владикавказский математический журнал 2014, Том 16, Выпуск 4, С. 61-64

УДК 519.17

THE EDGE CK GRAPH OF A GRAPH

P. Siva Kota Reddy, К. M. Nagaraja, V. M. Siddalingaswamy

For any integer k > 4, the edge Ck graph Ek (G) of a graph G = (V, E) has all edges of G as it vertices, two vertices in Ek (G) are adjacent if their corresponding edges in G are either incident or belongs to a copy of Ck. In this paper, we obtained the characterizations for the edge Ck graph of a graph G to be connected, complete, bipartite etc. It is also proved that the edge C4 graph has no forbidden subgraph characterization. Mereover, the dynamical behavior such as convergence, periodicity, mortality and touching number of Ek(G) are studied.

Mathematics Subject Classification (2000): 05C 99.

Key words: edge Ck graph, triangular line graph, line graph, convergent, periodic, mortal, transition number.

1. Introduction

For graph theory terminology and notation in this paper we follow the book [3]. All graphs considered in this paper are finite, unoriented, without loops and multiple edges.

Graph theory [3] is an established area of research in combinatorial mathematics. It is also one of the most active areas of mathematics that has found large number of applications in diverse areas including not only computer science, but also chemistry physics, biology anthropology psychology geography history economics, and many branches of engineering. Graph theory has been especially useful in computer science, since after all, any data structure can be represented by a graph. Furthermore, there are applications in networking, in the design of computer architectures, and in general, in virtually every branch of computer science. However, to date most of the research in graph theory has only considered graphs that remain static, i.e., they do not change with time. A wealth of such literature has been developed for static graph theory. Our purpose is to classify dynamic graphs, i.e., graphs that change with time. Dynamic graphs appear in almost all fields of science. This is especially true of computer science, where almost always the data structures (modeled as graphs) change as the program is executed. Very little is known about the properties of dynamic graphs.

The study of graph dynamics has been receiving wide attention, since Ore's work on the line graph operator L(G) (see [5, 6]). The edge Ck graph Ek(G) of a graph G is defined in [5] as follows: The edge Ck graph of a gr aph G = (V, E) is a graph Ek (G) = (V', E'), with vertex set V' = E(G) such that two vertices e and f are adjacent if, and only if, the corresponding

G Ck G

adjacent if, and only if, they belongs to a common P3 or Ck in G. When k = 3, the definition coincides with triangular line graph of a graph [2], and when k = 4, the definition coincides with E4-graph of a graph [4].

© 2014 Siva Kota Reddy P., Nagaraja К. M., Siddalingaswamy V. M.

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Siva. Kota Reddy P., Na.ga.raja K. M., Siddalingaswamy V. M.

Throughout this paper we denote by Pn (respectively Cn), a path (respectively cycle) on n vertices, The graph obtained by deleting any edge of Kn is denoted by Kn — e. A graph G is H-free if G does not contain H as an induced subgraph. A graph H is a forbidden subgraph

PP isomorphic to H. The cross product Gi x G2 of two graphs Gi and G2 is a simple graph with V(G1) x V(G2) as its vertex set and two vertices (u1; v1) and (u2, v2) are adjacent in G1 x G2 if, and only if, either u1 = u2 and v1 is adjacent to v2 in G2, or u1 is adjacent to u2 in G1 and v1 = v2.

Clearly, the edge Ck graph coincides with the line graph for any acyclic graph. But they differ in many properties. As a case, for a connected graph G, Ek (G) = G if, and only if, G = Cn, n = k. Also Beineke has proved in [1] that the line graph has nine forbidden subgraphs. In this paper, we see that Ek (G) has no forbidden subgraphs.

Ck

a graph G is connected, complete, bipartite etc. We have also proved that the edge C4 graph has no forbidden subgraph characterization. The dynamical behavior such as convergence, periodicity, mortality and touching number of Ek (G) are also studied.

Ck

The edge Ck graph Ek (G) of a graph G is defined in [5] as follows: The edge Ck graph of a graph G = (V, E) is a graph Ek(G) = (V', E'), with vertex set V' = E(G) such that two vertices e and f are adjacent if, and only if, the corresponding edges in G either incident or

Ck G

belongs to a common P3 or Ck in G. When k = 3, the definition coincides with triangular line graph of a graph [2], and when k = 4, the definition coincides with E4-graph of a graph [4].

Ck

in many properties. As a case, for a connected graph G, Ek (G) = G if and only if G = Cn, n = k. The following result characterizes graphs whose Ek graph is isomorphic to their line graph.

Theorem 1. For a graph G, Ek(G) = L(G) if, and only if, G is Ck-free.

Theorem 2. For any graph G, Ek(G) is connected if, and only if, exactly one component G

Theorem 3. For any graph G, the edge Ek graph is complete then diam(G) sC |_|J.

< Sinee Ek (G) is complete then by the definition of Ek (G) any two edges must either incident or belongs to a cycle of length k. Suppose that diam(G) > |_|J. That is there exists two vertices u and v in G with d(u, v) > |_|J. Clearly u and v can not be in the some cycle of length |_|J. Let v! and v' be any two vertices adjacent to u and v respectively. Then the edges uu' and vv' are not adjacent and does not belongs to a cycle of length k. This proves, a contradiction. >

Remark. The converse need not be true for example C5 has diameter 2 but E4(G) is not complete.

In [4], the authors prove that: For a connected graphG, E4(G) is complete if, and only if, G is complete multipartite. But the same can not be generalized for k ^ 5. For example for the peterson graph P, E5(P) is clearly complete graph K15. But P is not complete bipartite. However, we have the following:

Theorem 4. For a connected graph G, Ek (G) is complete if, and only if, every edge of G Ck

The Edge CK Graph of a. Graph

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Corollary 5. Let G be a complete r-multipartite graph for some r ^ Ek(G) is complete.

In [4], the authors proved that the edge C4 graph E4(G) has no forbidden subgraphs. We now prove that the edge Ck graph Ek (G) also have no forbidden subgraph characterization.

Theorem 6. There is no forbidden subgraph characterization for Ek(G) for any k ^ 3.

< We can assume that k ^ 5, since the edge C3 of a graph is nothing but triangular line graph and when k = 4, the result follows from the above result. We shall prove that given any graph G, we can find a graph H such that G is an induced subgraph of Ek(H). For anagraph G, let H = G x K2. Clearly H contains 2 copies of G say G and G'. Now let H' be the graph obtained from H by subdivide each edge of one copy of G in H into k — 4 edges.

We claim that G is induced subgraph of Ek(H'). For any v £ V(G), Ek(G) contains vertices of the form vv', where v' is the corresponding vertex in G'. Now for any two adjacent vertices u and v, the corresponding vertices uu' and vv' are also adjacent in Ek(G), since the vertices u, u', ui, u2, ■ ■ ■ , Uk-4, v', v forms a cycle of length k in H '.Now if u and v are non adjacent adjacent vertices of G then uu' and vv' are also non adjacent vertiees in Ek(H'). Thus the subgraph induced by the set {uu' : u £ G} of vertices in H'kG) contains G as a induced subgraph. This completes the proof. >

Theorem 7. For a connected graph G, Ek(G) is bipartite if, and only if, G is either a path or an even cycle of length r = k.

< Suppose that Ek(G) is bipartite. Suppose that G has a vertex of degree at least 3, then G contains a cycle of length 3. Hence, the degree of every vertex is at least 2. Since G connected, G must be a path or a cycle. Now, if G is an odd cycle of length r, then r can not be odd or equal to k. since, if r is odd then L(G) is also a cycle which is a subgraph of Ek(G) and if r = k, then Ek(G) = Kr and hence Ek(G) cannot be bipartite in both cases. Finally if r is even and r = k then Ek(G) = G, which is bipartite.

Conversely suppose that G is either a path or an even cycle of length r = k, then Ek (G) is either a path or an even cycle respectively. Hence Ek (G) is bipartite. >

Corollary 8. For a connected graph G, Ek (G) is a tree if, and only if, G is a path.

3. Dynamical Properties

G

n^-iterated graph is iteratively defined as follows: E0(G) = G, Ei(G) = Ek(G), En(G) = Ek(En-i(G)), n ^ 2. We say that G is convergent under Ek if {E^(G) : n £ N} is finite.

G Ek G Ek G

is some natural number n with G = Ejn(G). The smallest such number is called the period of G. The transition number t(x) of a convergent graph G is defined as zero if G is periodic and as the smallest number n such that En (G) is periodic. A graph G is mortal if for some n ^ N, En(G) = 0 the empty graph.

Theorem 9. The graphs Pn, K1>3, Cn (n = k) are the only Ek convergent graphs. < If G contains a vertex of degree > 3, then Ek(G) contains K4. In the subsequent

G

¿(G) < 3.

If G is a tree which, is neither Pn nor then K4 is contained at least in the third

iterated graph and hence G cannot converge. >

Corollary 10. For Ek(G), the only periodic graphs are the cycles Cn, n = k and they have period one.

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Siva. Kot a. Reddy P., Na.ga.ra.ja. К. M., Sidda.lingaswa.my V. M.

< The paths Pn converge to and K1>3 converges to the triangle. Consider the graphs which are not trees. If G is not a cycle, then G contains a cycle with a pendant edge as a subgraph (need not be induced). Then K4 is a subgraph at least in the second iteration and hence in the subsequent iterations the clique size will go on increasing and hence cannot converge. All cycles except Ck are fixed under E& and Ck is not convergent. Thus, the proof follows from

GG positive integer n with ЕП (G) periodic. >

Corollary 11. The transition number i(K1)3) = 1 and for n = k, t(Cn) = 0. Corollary 12. For Ek (G), the paths are the only mortal graphs.

< Among the convergent graphs, cycles other than Ck are fixed and K1>3 converges to K3.

The paths are the only graphs converging to ф >

References

1. Beineke L. W. Characterizations of derived graphs // J. Combinatorial Theory.—1970.—Vol. 9.—P. 129135.

2. Jarrett E. B. Transformations of graphs and digraphs. Ph.D. Thesis.—Western Michigan University, 1991.

3. Натягу F. Graph Theory.—Addison-Wesley Publ. Co., 1969.

4. Menon Manju K., Vijavakumar A. The edge C4 graph of a graph // Ramanujan Math. Soc. Proc. of ICDM (Bangalore, India, December 15-18, 2006).-2008.-P. 245-248.-(Lecture Notes Series, № 7).

5. Prisner E. Graph Dyanamics.—Longman, 1995.

6. Ore O. Theory of Graphs.—Providence (R. I.): Amer. Math. Soc., 1962,—Vol. 38.

Received January 15, 2014-

Six л Кота Reddy P. Department of Mathematics Siddaganga Institute of Technology, В. H. Road Tumkur-572 103, India

E-mail: [email protected]; pskreddyOsit. ac. in

Nagaraja К. M.

Department of Mathematics

J.S.S. Academy of Technical Education

Uttarahalli-Kengeri Main Road, Bangalore-560 060, India

E-mail: nagkmnOgmail. com

SlDDALINGASWAMY V. M.

Department of Mathematics J.S.S. Academy of Technical Education Uttarahalli-Kengeri Main Road, Bangalore-560 060, India E-mail: swamyvmsOyahoo. com

РЕБЕРНЫЙ Ck-ГРАФ ГРАФА Сива Кота Редди П., Нагарайя К. М., Сидцалингасвами В. М.

Для любого целого k > 4 реберный C^-гра ф Ek(G) граф a G содержит все ребра графа G в качестве вершин, при этом две вершины смежны в Ek(G), если соответствующие им ребра в графе G либо инцидентны, либо принадлежат копии Ck. В статье установлено, что реберный Ск-граф графа G является связным, полным, двудольным и т. д. Доказано также, что реберный С4-граф не имеет характеризаций запрещенными подграфами. Кроме того, исследованы такие характеристики дина-

Ek (G)

Ck

мортальность, число переходов.