Color the edges of the n-vertex complete graph red and blue, and suppose that red kcliques are fewer than blue k-cliques. We show that the number of red k-cliques is always less than ckn , where ck ∈ (0, 1) is the unique root of the equation z = (1 − z) + kz(1 − z)k−1. On the other hand, we construct a coloring in which there are at least ckn k −O(nk−1) red k-cliques and at least the same numbe...

In recent work (Pandit and Kulkarni [Discrete Applied Mathematics, 244 (2018), pp. 155–169]), the independence number of a graph was characterized as maximum \(\ell _1\) norm solutions Linear Complementarity Problem (LCP) defined suitably using parameters graph. Solutions this LCP have another relation, namely, that they corresponded to Nash equilibria public goods game. Thus \( \ell _1 \) has ...

Quasi-cliques are dense incomplete subgraphs of a graph that generalize the notion cliques. Enumerating quasi-cliques from is robust way to detect densely connected structures with applications in bioinformatics and social network analysis. However, enumerating challenging problem, even harder than problem We consider enumeration top- k degree-based make following contributions: (1) we show det...

Chimpanzees (Pan troglodytes) often groom in gatherings that cannot simply be divided into unilateral dyadic grooming interactions. This feature of grooming is studied at two different levels: grooming cliques and grooming clusters. Grooming cliques are defined as directly connected configurations of grooming interactions at any given moment, and when any member of a clique successively grooms ...

The realization graph G(d) of a degree sequence d is the whose vertices are labeled realizations d, where edges join that differ by swapping single pair edges. Barrus (2016) [3] characterized for which triangle-free. Here, any n≥4, we describe structure in exactly determines whether has clique size n. As consequence determine sequences complete on n vertices.

A clique clustering of a graph is a partitioning of its vertices into disjoint cliques. The quality of a clique clustering is measured by the total number of edges in its cliques. We consider the online variant of the clique clustering problem, where the vertices of the input graph arrive one at a time. At each step, the newly arrived vertex forms a singleton clique, and the algorithm can merge...

A strong clique in a graph is intersecting all inclusion-maximal stable sets. Strong cliques play an important role the study of perfect graphs. We class diamond-free graphs, from both structural and algorithmic points view. show that following five NP -hard or co-NP problems remain when restricted to graphs: Is given strong? Does have clique? every vertex contained Given partition set into cli...

In 1965, Motzkin and Straus related global maximaof a certain quadratic program to the maximum clique size in a certain graph. We extend this result to relate strict local maxima of this program to certain maximal cliques, and certain maxima to non cliques. Our results are useful to a companion paper which employs this QP in a neural net model to nd large cliques in graphs.

Two cliques of genes identified computationally for their high co-expression in the mouse brain according to the Allen Brain Atlas, and for their enrichment in genes related to autism spectrum disorder (ASD), have recently been shown to be highly co-expressed in the cerebellar cortex, compared to what could be expected by chance. Moreover, the expression of these cliques of genes is not homogen...

We consider the problem of generating all maximal cliques in an unit disk graph. General algorithms to find all maximal cliques are exponential, so we rely on a polynomial approximation. Our algorithm makes use of certain key geographic structures of these graphs. For each edge, we limit the set of vertices that may form cliques with this as the longest edge. We then consider several characteri...