Table of Contents
ISRN Combinatorics
Volume 2013, Article ID 703989, 4 pages
http://dx.doi.org/10.1155/2013/703989
Research Article

A Polynomial Representation and a Unique Code of a Simple Undirected Graph

1Department of Mathematics, Jadavpur University, Kolkata 700032, India
2Department of Pure Mathematics, University of Calcutta, Kolkata 700019, India

Received 30 June 2013; Accepted 29 July 2013

Academic Editors: E. Bannai, L. Clark, and D. S. Kim

Copyright © 2013 Shamik Ghosh et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We introduce a representation of simple undirected graphs in terms of polynomials and obtain a unique code for a simple undirected graph.

1. Introduction

Let be the set of all positive integers greater than 1. Let and be the set of all divisors of , greater than 1. Define a simple undirected graph with the vertex set and any two distinct vertices are adjacent if and only if . From an observation in [1], it follows that any simple undirected graph is isomorphic to an induced subgraph of for some . For the sake of completeness and further use of the construction we provide a sketch of the proof below. Throughout the paper by a graph, we mean a simple undirected graph.

Theorem 1. Let be a graph. Then is isomorphic to an induced subgraph of for some .

Proof. Let be a graph. Let be the set of all maximal cliques of . For , let be the th prime. For each , define . Now in order to make the values of distinct for distinct vertices, we modify by using different powers of primes , if required. For each , let be the modified value of . Let be the least common multiple of . Now it is clear that for any ,  Thus, is isomorphic to the subgraph of induced by the set of vertices of .

Example 2. Consider the graph in Figure 1. The maximal cliques of are , , , , and . We assign the th prime to the clique for . Then for each , we compute and as in the proof of Theorem 1 (Table 1).
So is isomorphic to the subgraph of induced by the set of vertices of , where .

tab1
Table 1
703989.fig.001
Figure 1: The graph in Example 2.

2. Main Results

It is important to note that instead of taking all maximal cliques of in Theorem 1, it is sufficient to consider a set of cliques of which covers both vertices and edges of .

Definition 3. Let be a graph. A set of cliques of is called a total clique covering of if covers both and . The minimum size of a total clique covering of is called the total clique covering number of . We denote it by .

On the other hand, given a finite sequence of positive integers, one can construct a simple undirected graph as follows.

Definition 4. Let be a finite sequence of positive integers. Then corresponding to this sequence, define a graph , where , corresponds to ( is called the label of ) for , and if and only if and . The graph is said to be realized by the sequence .

Now by Theorem 1, every graph can be realized by some sequence of positive integers. Also in Definition 4, it is sufficient to take the entries of square-free if . Thus, for convenience, we specify the sequence in the following manner.

Definition 5. Let be a graph. Let , and let the graph be obtained from by deleting isolated vertices of , if there are any. Let be a finite nondecreasing sequence of square-free positive integers greater than 1 such that . If has isolated vertices, then we prefix number of 1’s in to obtain the sequence . For , . If is a graph with a single vertex, then . Then and the sequence is called a coding sequence of . The entries of , which are greater than , are called nontrivial. Let be the least common multiple of all nontrivial entries of .

Lemma 6. Let be a coding sequence of a graph . Let , where corresponds to for all (one writes ). Let be the set of all distinct prime factors of . Suppose has isolated vertices. Define for and for each , define . Then is a total clique covering of such that , and for all if and only if and for each , if and only if .

Proof. We first note that is a clique containing only the vertex for . Now for each , divides for some as divides . Thus, . So , for all . Also for any two vertices , divides both and . So and are adjacent in . Thus, each is a clique of . Again since is the set of all prime factors of , each vertex belongs to for some . So covers . Moreover, two vertices and are adjacent if and only if . So and must have at least one common prime factor, say, and hence and both lie in . So also covers and hence is a total clique covering of . Now (, ) and for all if and only if divides but does not divide for all . Now since nontrivial entries of are square-free integers, is a product of distinct primes and hence . By Definition 5, for each , if and only if .

Definition 7. Let be a coding sequence of a graph . Then the total clique covering of as defined in Lemma 6 is called the total clique covering of corresponding to the sequence and is denoted by .

Lemma 8. Let be a graph, and let be a total clique covering of . Then there exists a coding sequence of such that .

Proof. Let , where for , . Let denote the th prime. We define a map by if and if (where ) and for all . Now being a total clique covering, every vertex is assigned a label in this way. Also note that two vertices , are adjacent in if and only if they lie in some and consequently, if and only if and have a common prime factor . Let us arrange the vertices of according to the nondecreasing order of and define the sequence by , where . Then clearly is a coding sequence of and .

Theorem 9. Let be a graph with isolated vertices, and let be the minimum number of prime factors of among all coding sequences of . Then .

Proof. Let be a coding sequence of such that the number of prime factors of is . Then by Lemma 6, . Again let be a total clique covering of such that . Then by Lemma 8, there is a coding sequence of such that and it follows from the proof of Lemma 8 that there are prime factors of . Thus, . Therefore, .

Definition 10. Let be a graph with and isolated vertices. Let . Let be a total clique covering of such that and for . Now as in the proof of Lemma 8, there are ways of assigning the first primes to the cliques to obtain at most different coding sequences. Let be the least among them in the lexicographic ordering in . Then is called the coding sequence of with respect to . Let be the set of all total clique coverings of such that for all . Let be the least element of in the lexicographic ordering in . Then is called the code of the graph .

For example, and are two coding sequences of the graph in Example 2 with respect to the total clique covering . It is easy to see that and is the only total clique covering with 5 cliques. The code of is . Note that for any graph , we have .

Theorem 11. Let and be two graphs. Then if and only if .

Proof. The proof follows from Definition 10.

Definition 12. Let be a graph with isolated vertices. Let be a total clique covering of , where for . If , let be a monomial in the polynomial semiring of indeterminates over the semiring of nonnegative integers with usual addition and multiplication. For any , , define . Now we define Then is said to be a polynomial representation of with respect to .

Consider the graph in Example 2. Then is a polynomial representation of with respect to . Now let be a graph, and let be a coding sequence of . From the construction of (cf. Lemma 6, Definition 7) and by Definition 12, it follows that can also be obtained from by replacing primes by () and commas by the addition symbol. It is important to note that the constant term of in Definition 12 is the number of isolated vertices of .

Definition 13. Let be a graph and the code of . Then the polynomial representation of corresponding to , that is, , is called the normal polynomial representation or the canonical polynomial representation of and is denoted by .

The normal polynomial representation of the graph in Example 2 is given by The following interesting observations are immediate from Definition 13.

Observation 1. A graph is disconnected if and only if , where and are polynomials with no common variables between them. The same is true for for any total clique covering of .

Observation 2. A graph is bipartite if and only if , where monomials belonging to the same have no common variables, for . The same is true for for any total clique covering of .

We now proceed to obtain a formula for .

Theorem 14. Let , where the ’s are distinct primes. Then and there is only one total clique covering of with precisely cliques.

Proof. If is prime, then the result is obvious. Suppose is not a prime number. Now the vertices of correspond to all the divisors of , greater than 1. So there are vertices labeled , . Clearly, no two of them can lie in the same clique as for all . So in any total clique covering of , these vertices will be in different cliques. In other words, any total clique covering has at least cliques. Now let . Then is easily seen to be a total clique covering of . So we have a total clique covering consisting of precisely cliques. Hence, .
Now we show that there is only one total clique covering of containing precisely cliques. Suppose there is another total clique covering of with exactly cliques. Here also, the vertices labeled , , are in distinct cliques. Without loss of generality, let for . Now for any , is not adjacent to those vertices which are not in . So for all . Now suppose that for some , there is a vertex in which is not in . Now being a total clique covering, has to lie in at least one , where . Since a total clique covering covers all the edges, the edge between and will be covered. implies that and both lie in some , where . However, this contradicts the fact that is not adjacent to . So for all and hence .

Let denote the maximum size of an independent set in a graph . In general, for any graph . The proof of the following proposition is similar to that of Theorem 14 and so it is omitted.

Proposition 15. Let be a graph with . If is an independent set and is a total clique covering of such that each lies only in among cliques in for each , then , and is the only total clique covering containing exactly cliques.

Now we provide a formula for .

Theorem 16. Let , where the ’s are distinct primes and , . Then contains all the monomials , where , with the coefficient , that is, unless is prime. If is prime, then .

Proof. If is prime, then and so . Suppose is not prime. Now by Theorem 14, and there is only one total clique covering, say, of with precisely cliques. Let , where for . So is . First, let . Now is the least (in the lexicographic ordering) among the coding sequences of obtained from . By Definition 10, any such coding sequence involves the primes , where is the prime. Now for any , there are elements (namely, ) which belong to only the clique . So considering the labellings and the fact that , it is easy to see that the coding sequence will be the least in the lexicographic ordering if we assign to the clique . So the vertex corresponding to a number ( for where ) is assigned the label . In other words, to find out , we first consider the set of all divisors (say) of , greater than 1, where . Then in each , we first replace by , then make the resultant entries square-free and arrange them in nondecreasing order. This gives us , where each is a product of primes of the form , () and this particular number repeats, say, times in the sequence , where is the number of divisors of (greater than 1) which are of the form , for ; that is, .
Thus, by Definition 13 we have
Finally, if , then . So , which satisfies the aforementioned formula. This completes the proof.

For example, , where , , and are distinct primes. Further, one may easily verify codes and normal polynomial representations for the following special classes of graphs:(i), , ( times), for , (ii), for , (iii), for , where is the prime for and , , and are, respectively, the complete graph, the path, and the cycle with vertices.

3. Conclusion

There are some representations of simple undirected graphs in terms of adjacency matrices, adjacency lists, and unordered pairs which are not unique for isomorphic graphs. There are some other instances for unique representations [2, 3]. In the nauty algorithm, McKay [2] defined a canonical isomorph [4] which is a graph rather than a sequence of integers as the code of a graph introduced here. The importance of the code is its uniqueness and its simple form. It is the same for any set of isomorphic graphs. The determination of is not always easy, but once it is obtained for a graph, it becomes the characteristic of the graph. The authors believe that further study of the code and the normal polynomial representation of a simple undirected graph will be helpful in further research on graph theory. The purpose of this paper is to communicate these interesting observations to all graph theorists.

Acknowledgment

The authors gratefully acknowledge the learned referees for their kind suggestions and comments which enriched the paper.

References

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