Some More Results on Facto Graphs

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1 Int. Journal of Math. Analysis, Vol. 6, 2012, no. 50, Some More Results on Facto Graphs E. Giftin Vedha Merly Scott Christian College (Autonomous), Nagercoil Kanyakumari District, Tamil Nadu, India N. Gnanadhas Former Principal, Nesamony Memorial Christian College, Marthandam, Kanyakumari District, Tamil Nadu, India Abstract In this paper we have extended the concept of Facto Graph which was initiated in [1]. A facto graph of integral order z, (which is a positive integer) is a graph G with the set of divisors(factors) of z as its vertex set and two distinct vertices in G are adjacent if the product of the two numbers which are labeled to the vertices is in the vertex set of G. Here we contribute some new results related to the particular case of facto graph, called Perfect Facto graph. Keywords: Facto graph, Perfect facto graph, Integral order 1. INTRODUCTION By a graph we mean a finite, undirected, non-trivial, connected graph without loops and multiple edges. The order and size of a graph are denoted by p and q respectively. For terms not defined here we refer to Frank Harary [3].

2 2484 E. Giftin Vedha Merly and N. Gnanadhas For any positive integer z, we define the facto graph as G(z) = (V, E) where V = {v i / v i is a factor of z} and two vertices v i and v j are adjacent if and only if v i v j V. A graph G is said to be a facto graph if there exists a positive integer z such that G is isomorphic to a facto graph G(z) for some z. If G G(z) for some z, then the integral order of the graph G is equal to z and is denoted by o i (G). For simplicity we refer a facto graph G(z) by G. A clique of a graph G is a complete subgraph of G. A clique of G is a maximal clique if it is not properly contained in another clique of G. the order of the maximal clique of g is called the clique number and is denoted by ω(g). For v V, d(v) is the number of edges incident with v. In [1] we have defined the facto graph, its integral order and also the order and size of particular type of facto graph have been obtained. In this paper we attempt to study more on particular type of facto graph namely perfect facto graph. In particular the degree sequence and the clique number of each type are obtained and also some of the results related are dealt with. 2. PERFECT FACTO GRAPH Definition 2.1: A facto graph G with o i (G) = α where is a prime and α is a positive integer is called a perfect facto graph. If α is odd (even), then G is called an odd (even) perfect facto graph. First we concentrate on the degree sequence of the perfect facto graphs. For z = 8, consider the corresponding facto graph G which is depicted in figure 1.

3 Some more results on facto graphs 2485 Figure 1 From the graph G we observe that two vertices i and j are adjacent whenever the sum of the powers of i and j is less than or equal to 8. And the degree sequence of G is 8, 7, 6, 5, 4, 4, 3, 2,1. Theorem 2.2: If α is a positive even integer and is any prime, then G is an even perfect facto graph with o i (G) = α if and only if G is of order α + 1 and the degree sequence of G is given by,1,2,, 1,,, 1,,2,1, where d = d =. Proof: Let G = (V, E) be an even perfect facto graph with o i (G) = α. We have the number of positive divisors of α is α + 1 and hence the order of G is α + 1. Let V = { 0, 1, 2, 3,, α }be the vertex set of G. In a perfect facto graph G, we observe that for i j, the edge i j is in G if and only if i + j α. Thus

4 2486 E. Giftin Vedha Merly and N. Gnanadhas the vertex 0 is adjacent to the vertices 1, 2, 3,, α and d( 0 ) = α. The vertex 1 is adjacent to the vertices 0, 2, 3,, α-1 and d( 1 ) = α 1. The vertex 2 is adjacent to the vertices 0, 1, 3,, α-2 Thus d( 2 ) = α 2. Similarly d( 3 ) = α 3. Continuing this way we have the vertex which is adjacent to the vertices 0, 1, 2,,, and so that 1. The vertex is incident with the edges p 0, 1, 2,, so that. The vertex is incident with the edges 0, 1, 2,,. Hence. The vertex is incident with the edges 0, 1, 2,,. Therefore 1. Now it is clear that, the successive vertices will have degree one less than that of the previous vertex and the degree of the vertex α-1 is 2 and that of α is 1. That is d( 0 ) = α, d( 1 ) = α 1, d( 2 ) = α 2,, 1,, 1,, d(α-1 ) = 2 and d( α ) = 1.Hence the degree sequence of G is s:,1,2,, 1,,, 1,,2,1. Conversely suppose that G is of order α + 1 and the degree sequence of G is,1,2,, 1,,, 1,,2,1. It is possible to construct a graph G with order α + 1 and is a realization of and let α, α 1, α 2,, 1,,, 1,, 2, 1 are the degrees of the vertices 0, 1, 2,,,, α-1, α respectively. That is d( i ) = α i if 0 i and d(j ) = α (j 1) if + 1 j α and the graph with this condition satisfied is an even perfect facto graph of integral order α.hence G is an even perfect facto graph of integral order α. Theorem 2.3: If α is an odd positive integer and is any prime, then G is an odd perfect facto graph with o i (G) = α if and only if G is of order α + 1 and the degree sequence of G is given by :,1,2,, 1,,, 1,,2,1 where. Proof: Let G = (V, E) be an odd perfect facto graph with o i (G) = α. We have the number of positive divisors of α is α + 1 and hence the order of G is α + 1. Let V =

5 Some more results on facto graphs 2487 { 0, 1, 2, 3,, α }. As proved in Theorem 2.2, we have d( 0 ) = α, d( 1 ) = α 1, d( 2 ) = α 2, and instead of we have in this graph. The vertex is adjacent to the vertices 0, 1, 2,, and. Thus 1. The vertex is adjacent to the vertices 0, 1, 2,, and so that. Also we have is adjacent to 0, 1, 2,, and hence and the successive vertices will have degree one less than that of the previous so that d( α-1 ) = 2 and d( α ) = 1. Thus the degree sequence of G is given by,1,2,, 1,,, 1,,2,1 with. Theorem 2.4: If G is a perfect facto graph with o i (G) = α, then where o i ( ) = α-1. Proof: Let G = (V, E) be a perfect facto graph with o i (G) = α. Then the order p of G is + 1 and each vertex of G may be incident with atmost edges. Let V = { 0, 1, 2, 3,, α } be the vertex set of G. Case (i): is even. The degree sequence of G is given by,1,2,, 1,,, 1,,2,1 where d( i ) = α i if 0 i and d(j ) = α (j 1) if + 1 j α. Then the complement of G has the degree sequence, say :,1, 2,, 1,,, 1,,2,1. That is :0,1,2,, 1,,,1,,2,1. The rearrangement of is given by :,1,2,, 1,,, 1,,2,1,0. Now let α1. Then s :, 1, 2,, +1,,,, 2,1,0. Then by Theorem 2.3, we get s is the degree sequence of an odd perfect facto graph of integral order p α and union of an isolated vertex. Hence where.

6 2488 E. Giftin Vedha Merly and N. Gnanadhas Case (ii): is odd. As in the previous case the complement of G will have the degree sequence of an even perfect facto graph of integral order p α-1 and a union of an isolated vertex so that with. Theorem 2.5: The clique number of an even perfect facto graph G of integral order α is 1. Proof: Let G = (V, E) be an even perfect facto graph and o i (G) = α and let V = { 0, 1, 2, 3,, α } be the vertex set of G and we consider the set S which consist of the first vertices of V. That is, S = { 0, 1, 2, 3,, }. We try to prove that the subgraph of G induced by S is the maximal clique of G. Claim: is a clique of G. In a perfect facto graph G, we observe that for i j, the edge i j is in G if and only if i + j α. Now we consider each vertex of S in G. The vertex 0 is adjacent to all the other vertices of G so that to all the vertices of S. The vertex 1 is adjacent to all the vertices of G other than α and hence adjacent to all the vertices of S. Continuing this way we have the vertex is adjacent to the vertices 0, 1, 2, 3,,, and.thus it is adjacent to all the vertices of S other than itself. Also it is clear that the vertex is adjacent to 0, 1, 2, 3,,, that is all the vertices of S other than itself, but not adjacent to. Thus the subgraph induced by S is a complete graph. Hence the claim holds. It remains to prove that is the maximal clique of G. For, we consider the next vertex to in G, in the order they are arranged in V. That is, the vertex. It is adjacent to the vertices 0, 1, 2, 3,, but it is not adjacent to of S. Also we have d( i ) = α (i 1) if + 1 i α so that d(j ) 1 j α. Hence it is not possible to have a clique which is larger than the subgraph induced by S. Therefore is the largest clique of G and hence the clique number of G is given by ω(g) = 1.

7 Some more results on facto graphs 2489 Theorem 2.7: Every odd perfect facto graph G of integral order α has clique number, ω(g) = 1. Proof: Let G = (V, E) be a facto graph of integral order α where is an odd positive integer and let V = { 0, 1, 2, 3,, α }. It is to prove that an induced subgraph of G is a complete graph of order. Now let S = { 0, 1, 2, 3,, }. Claim: is a clique of G. It is observed that two distinct vertices i and j in the graph G are adjacent if and only if i + j α. Now for j = 1,2, 3, 4,, and i = 0,1, 2, 3,, with i j, let the value of i j = k. The maximum value of k occurs when j = and i and the value of k =. The edge set given by { i j / j = 1,2, 3, 4,,, i = 0, 1, 2, 3,, and i j} is exactly the edge set of. Thus the vertices in S are mutually adjacent and hence the subgraph of G induced by S is a complete graph. Hence the claim holds. It remains to prove that the subgraph is the largest clique of G. That is to prove that the subgraph induced by S is the maximum clique of G. For, we consider the next vertex of in G in the order they are arranged in V. That is, the vertex. It is adjacent to the vertices 0, 1, 2, 3,,, but not adjacent to of S. Also for an odd perfect facto graph we have d( i ) = α (i 1) if i α so that d( j ) < for j α. Hence it is not possible to have a clique which is larger than the subgraph induced by S. Therefore is the maximum clique of G and hence the clique number of G is ω(g) = +1. Theorem 2.9: If G is a perfect facto graph and o i (G) = α where is an odd positive integer and is a prime, then \, if α is even and \, if α is odd.

8 2490 E. Giftin Vedha Merly and N. Gnanadhas Proof: G is a perfect facto graph and o i (G) = α. Let K α+1 be a complete graph of order α + 1 and let the vertex set of K α+1 be {v 0, v 1, v 2,, v α }. Case (i): α is even. In K α+1 remove the edge set E(K 1,α-1 ) ={ v α v 1, v α v α v 2,..., v α v α- v α-1 }, which are α 1 in number and are incident with v α. The degree sequence of the resulting graph is α, α 1, α 1,, α 1, 1. Now remove the edge set E(K 1,α-3 ),the edges incident with v α-1, namely v α-1 v 2, v α-1 v 3,..., v α-1 v α-2 and are α -3 in number and the degree sequence of the resulting graph is α, α -1, α -2,, α-2,α- 2,2,1. Since α is even, after steps of removal of the edges, we have E(K 1,1), (i.e., the edge ), and and are of same degree. And the degree sequence of the resulting spanning subgraph \, is α, α -1, α - 2,..., +1,,, -1,, 2, 1. By theorem 2.2 we have the graph \, is a facto graph of integral order α, α is even. Thus we obtain \,. Case (ii): α is odd. In the complete graph K α+1, for i = 1, 2,, we remove the edge sets E(K 1,2i ) in successive steps, as in previous case and since α is odd, after steps we will have the vertices and each of degree +1. And the degree sequence of the spanning subgraph \, of is α, α 1, α 2,, 1, 1,, 2, 1. By Theorem 2.3, it is clear that the +2, 1,, perfect facto graph \,. Theorem 2.11: If G is a perfect facto graph of integral order α, then where.

9 Some more results on facto graphs 2491 Proof: Let G = (V, E) be a perfect facto graph with (G) = α. Then the order p of G is +1 and each vertex of G may be incident with atmost edges. Let V = { 0, 1, 2, 3,, α } be the vertex set of G. Case (i): is even. The degree sequence of G is given by :, -1, α-2,..., 1,,, 1,..., 2, 1 where d(i ) = α i if 0 i 1, = and d( j ) = α (j 1) if + 2 j α. Then the complement of G has the degree sequence, say :, ( 1), ( 2),..., 1,,, 1,..., 2, 1. That is : 0, 1, 2,, 1,,, 1,, 2, 1. The rearrangement of is given by : -1, α-2,, 1,,, 1,, 2,1, 0. Now let 1. Then :, 1, 2,, +1,,,..., 2,1,0. By Theorem 2.3 we get s is the degree sequence of the odd perfect facto graph of integral order and a union of an isolated vertex. Hence where. Case (ii): is odd. As in the previous case the complement of G will have the degree sequence of an even perfect facto graph of integral order p α-1 and a union of an isolated vertex so that with. References [1] E.Giftin Vedha Merly and N.Gnanadhas, On Facto Graph, International Journal of Mathematics Research, Volume 4,Number 2(2012),pp [2] Elementary Number Theory - David M. Burton, University of New Hampshire. [3] Frank Harary, 1872, Graph Theory, Addition-Wesley Publishing Company. [4] Zhibo Chen, Integral Sum Graphs from identification, Discrete Math. 181 (1998)

10 2492 E. Giftin Vedha Merly and N. Gnanadhas Received: April, 2012

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