Maximal Cliques in Graphs: Exploring the Number of Distinct Sizes

Graph theory is a fascinating branch of mathematics that provides insight into complex network structures. One of the intriguing problems in graph theory involves the study of cliques, which are fully connected subgraphs. Specifically, we are interested in the number of distinct sizes of maximal cliques for a graph with N vertices.

Introduction to Maximal Cliques

A clique in a graph is a subset of vertices where every two distinct vertices are adjacent. A clique is considered maximal if it is not a subset of any other clique in the graph. The study of these maximal cliques can help us understand the connectivity and structure of a graph.

Understanding the Problem

The query we are addressing is: What is the maximum number of distinct sizes of maximal cliques in a graph with N vertices?

Initial Examples

Let's start with some basic examples to understand the concept:

N 3: The answer is 2 because we can have a vertex and an edge {2,3}. If we had all vertices K_3, we could only have a single clique. N 4: Similarly, we cannot get 1, 2, and 3 sizes. If we had a single vertex, it would be disjoint from the other three. Hence, the answer remains 2.

General Case Analysis

Now, let's consider a graph with n vertices and maximal cliques with p different sizes, none of which are singletons. We can denote the graph by A_n and its maximal cliques by S_n. We then create a new graph B_{n,1} by removing a vertex from A_n, denoted as n,1. The number of distinct maximal clique sizes in B_{n,1} is denoted as p_1. Given B_{n,1}, we can create A_{n,1} by adding n,1 to each clique in B_{n,1}. In this case, the number of distinct sizes does not increase.

This process shows that we can get p_n to grow like n/2. However, it is not clear if we can achieve a higher number.

Computational Evidence

Recent computational studies have revealed that for n 7, there is a second solution with no singleton cliques, which produces a larger solution for n 8 and beyond. This suggests that the answer may be more complex than initially thought.

Revised Conjecture

Further computer output indicates that if d n - p represents the deficit between n and the number of distinct maximal clique sizes, then it seems that d grows like the ceiling of -1 * sqrt(n) - 7/2.

Application and Connection to OEIS

While the problem is not specified to be on a connected graph, the answer lies in the largest number of distinct positive integers whose sum is N. This query can be found in the OEIS - the Online Encyclopedia of Integer Sequences. Here are some examples:

NPossible Distinct Sizes 11, 1 21, 2 32, 1, 2 42, 1, 3 52, 1, 4, 2, 3 63, 1, 2, 3 73, 1, 2, 4 81, 1, 2, 5 92, 1, 2, 6, 2, 3, 4 101, 1, 2, 3, 4

Understanding these examples and patterns can provide valuable insights into the structure and properties of graphs with various sizes of maximal cliques.

Conclusion

The problem of finding the maximum number of distinct sizes of maximal cliques in a graph with N vertices remains an open question with interesting computational and theoretical implications. Future research may reveal new insights and further refine our understanding of graph cliques.

The study of maximal cliques not only enriches our understanding of graph theory but also has applications in various fields, including social networks, computer networks, and data mining.