Related papers: Clique complexes and graph powers
In this article, we introduce the notion of a wedge of graphs and provide detailed computations for the independence complex of a wedge of path and cycle graphs. In particular, we show that these complexes are either contractible or wedges…
A strong clique in a graph is a clique intersecting every maximal independent set. We study the computational complexity of six algorithmic decision problems related to strong cliques in graphs and almost completely determine their…
It is shown how to construct a clique graph in which properties of cliques of a fixed order in a given graph are represented by vertices in a weighted graph. Various definitions and motivations for these weights are given. The detection of…
We use two cofibre sequences to identify some combinatorial situations when the independence complex of a graph splits into a wedge sum of smaller independence complexes. Our main application is to give a recursive relation for the homotopy…
We study cliques in graphs arising from quadratic forms where the vertices are the elements of the module of the quadratic form and two vertices are adjacent if their difference represents some fixed scalar. We determine structural…
We give some new bounds for the clique and independence numbers of a graph in terms of its eigenvalues.
In this paper, we present examples of the cyclic sieving phenomenon coming from studying independent sets in graphs of a fixed size k. Given a graph G, and a cyclic group C acting on the graph, then C also acts on the collection of…
The ordinary generating function of the number of complete subgraphs (cliques) of $G$, denoted by $C(G,x)$, is called the The clique polynomial of the graph $G$. In this paper, we first introduce some \emph{clique} incidence matrices…
We propose and study a hierarchical algorithm to generate graphs having a predetermined distribution of cliques, the fully connected subgraphs. The construction mechanism may be either random or incorporate preferential attachment. We…
This paper proposes a new algorithm for solving maximal cliques for simple undirected graphs using the theory of prime numbers. A novel approach using prime numbers is used to find cliques and ends with a discussion of the algorithm.
Counts of small subgraphs, or graphlet counts, are widely applicable to measure graph similarity. Computing graphlet counts can be computationally expensive and may pose obstacles in network analysis. We study the role of cliques in…
Scale-free networks contain many small cliques and cycles. We model such networks as inhomogeneous random graphs with regularly varying infinite-variance weights. For these models, the number of cliques and cycles have exact integral…
We consider a set of cliques in any multipartite graph with two vertices in each part. Moreover, we construct a class of peculiar polytopes. Key words: multipartite graph, clique, polytope.
The clique complex of a graph G is a simplicial complex whose simplices are all the cliques of G, and the line graph L(G) of G is a graph whose vertices are the edges of G and the edges of L(G) are incident edges of G. In this article, we…
In this note, we describe a construction that leads to families of graphs whose critical groups are cyclic. For some of these families we are able to give a formula for the number of spanning trees of the graph, which then determines the…
It is known that non-isomorphic strongly regular graphs with the same parameters must be cospectral (have the same eigenvalues). In this paper, we investigate whether the spectra of higher order Laplacians associated with these graphs can…
We study graphon counterparts of the chromatic and the clique number, the fractional chromatic number, the b-chromatic number, and the fractional clique number. We establish some basic properties of the independence set polytope in the…
Clique-width is a well-known graph parameter. Many NP-hard graph problems admit polynomial-time solutions when restricted to graphs of bounded clique-width. The same holds for NLC-width. In this paper we study the behavior of clique-width…
Let $G$ be a group. The power graph of $G$ is a graph with vertex set $G$ in which two distinct elements $x,y$ are adjacent if one of them is a power of the other. We characterize all groups whose power graphs have finite independence…
We present a number of relations involving the number of cliques in a graph and its spectral radius.