Our primary objective in this paper is to study the distribution of the maximal clique size of the vertices in complex networks. We define the maximal clique size for a vertex as the maximum size of the clique that the vertex is part of and such a clique need not be the maximum size clique for the entire network. We determine the maximal clique size of the vertices using a modified version of a...

Clique problem has a wide range of applications due to its pattern matching ability. There are various formulations clique like k-clique problem, maximum etc. The k-Clique determines whether an arbitrary network or not whereas finds the largest in graph. It is already exhibited literature that (NP-problem) can be solved asymptotically faster by using quantum algorithms compared conventional com...

The clique tree algorithm is the standard method for doing inference in Bayesian networks. It works by manipulating clique potentials — distributions over the variables in a clique. While this approach works well for many networks, it is limited by the need to maintain an exact representation of the clique potentials. This paper presents a new unified approach that combines approximate inferenc...

let $g$ be a non-abelian group and let $z(g)$ be the center of $g$. associate with $g$ there is agraph $gamma_g$ as follows: take $gsetminus z(g)$ as vertices of$gamma_g$ and joint two distinct vertices $x$ and $y$ whenever$yxneq yx$. $gamma_g$ is called the non-commuting graph of $g$. in recent years many interesting works have been done in non-commutative graph of groups. computing the clique...

Let C be a family of cliques of a graph G = (V,E). Suppose that each clique C of C is associated with an integer r(C), where r(C) ≥ 0. A vertex v r-dominates a clique C of G if d(v, x) ≤ r(C) for all x ∈ C, where d(v, x) is the standard graph distance. A subset D ⊆ V is a clique r-dominating set of G if for every clique C ∈ C there is a vertex u ∈ D which r-dominates C. A clique r-packing set i...

Clique-width of graphs is a major new concept with respect to efficiency of graph algorithms. The notion of clique-width extends the one of treewidth, since bounded treewidth implies bounded clique-width. We give a complete classification of all graph classes defined by forbidden induced subgraphs of at most four vertices with respect to bounded or unbounded clique-width.

Given positive integers m, n, we consider the graphs Gn and Gm,n whose simplicial complexes of complete subgraphs are the well-known matching complex Mn and chessboard complex Mm,n. Those are the matching and chessboard graphs. We determine which matching and chessboard graphs are clique-Helly. If the parameters are small enough, we show that these graphs (even if not clique-Helly) are homotopy...

A clique in a graph is a set of vertices, each of which is adjacent to every other vertex in this set. A k-clique relaxes this requirement, requiring vertices to be within a distance k of each other, rather than directly adjacent. In theory, a maximum clique algorithm can easily be adapted to solve the maximum k-clique problem, although large sparse k-clique graphs reduce to large dense clique ...