Given an undirected graph G, let PG(z) be the polynomial PG(z) = ∑ n (−1)ncnz , where cn is the number of cliques of size n in G. We show that, for every G, the polynomial PG(z) has only one root of smallest modulus. Clique polynomials are related to trace monoids. Indeed, 1 PG(z) is the generating function of the sequence {tn}, where tn is the number of traces of size n in the trace monoid def...

Abstract. An efficient algorithm is presented to compute the characteristic polynomial of a threshold graph. Threshold graphs were introduced by Chvátal and Hammer, as well as by Henderson and Zalcstein in 1977. A threshold graph is obtained from a one vertex graph by repeatedly adding either an isolated vertex or a dominating vertex, which is a vertex adjacent to all the other vertices. Thresh...

In this paper, we propose a new approach for solving the SAT problem. This approach consists in representing SAT instances thanks to an undirected graph issued from a polynomial transformation from SAT to the CLIQUE problem. Considering this graph, we exploit well known properties of chordal graphs to manipulate the SAT instance. Firstly, these properties allow us to define a new class of SAT p...

Let T = (V,A) be a directed tree. Given a collection P of dipaths on T , we can look at the arc-intersection graph I(P, T ) whose vertex set is P and where two vertices are connected by an edge if the corresponding dipaths share a common arc. Monma and Wei, who started their study in a seminal paper on intersection graphs of paths on a tree, called them DE graphs (for directed edge path graphs)...

This paper studies the minimax detection of a small submatrix of elevated mean in a p× p matrix contaminated by additive Gaussian noise where p tends to infinity. To investigate the tradeoff between statistical performance and computational cost from a complexity-theoretic perspective, we consider a sequence of discretized models which are asymptotically equivalent in the sense of Le Cam. Under...

Ray intersection graphs are intersection graphs of rays, or halflines, in the plane. We show that any planar graph has an even subdivision whose complement is a ray intersection graph. The construction can be done in polynomial time and implies that finding a maximum clique in a segment intersection graph is NP-hard. This solves a 21-year old open problem posed by Kratochv́ıl and Nešetřil. ∗Rese...

We are given a graph G with n vertices, where a random subset of k vertices has been made into a clique, and the remaining edges are chosen independently with probability 1 2 . This random graph model is denoted G(n, 1 2 , k). The hidden clique problem is to design an algorithm that finds the k-clique in polynomial time with high probability. An algorithm due to Alon, Krivelevich and Sudakov us...

In this paper, I submit an algorithm that efficiently solves the problems k-clique, maximum clique, and complete enumeration of all maximal cliques for any given graph that I have written in the Java programming language. I will first give a high level overview of how the cliques are calculated using this algorithm and show a more in depth look as to how my algorithm works as well as present th...

It is shown that the Hilbert series of the face ring of a clique complex (equivalently, flag complex) of a graph G is, up to a factor, just a specialization of S G (x, y), the subgraph polynomial of the complement of G. We also find a simple relationship between the size of a minimum vertex cover of a graph G and its subgraph polynomial. This yields a formula for the h-vector of the flag comple...

Cliques are commonly used for social network analysis tasks, as they are a good representation of close-knit groups of people. For this reason (as well as for others), the problem of enumerating, i.e., finding, all maximal cliques in a graph has received extensive treatment. However, considering only complete subgraphs is too restrictive in many real-life scenarios where “almost cliques” may be...