Let G be a graph and KG be the set of all cliques of G, then the clique graph of G denoted by K(G) is the graph with vertex set KG and two elements Qi, Qj ∈ KG form an edge if and only if Qi ∩ Qj 6= ∅. Iterated clique graphs are defined by K0(G) = G, and K(G) = K(Kn−1(G)) for n > 0. In this paper we determine the number of cliques in K(G) when G = G1+G2, prove a necessary and sufficient conditi...

In this paper we present an exact algorithm for the maximum-weight clique problem on arbitrary undirected graphs. The algorithm based on a fact that vertices from the same independent set couldn’t be included into the same maximum clique. Those independent sets are obtained from a heuristic vertex coloring where each of them is a color class. Color classes and a backtrack search are used for pr...

Cheng Zhao∗, Yuejian Peng, Indiana State University In 1965, Motzkin and Straus establishes a remarkable connection between the maximum clique problem and the extrema of the Lagrangian of a graph. It is interesting to study a generalization of the Motzkin-Straus Theorem to hypergraphs. In this paper, we give some Motzkin-Straus type results.

Computing the clique number and chromatic number of a general graph are well-known to be NP-Hard problems. Codenotti et al. (Bruno Codenotti, Ivan Gerace, and Sebastiano Vigna. Hardness results and spectral techniques for combinatorial problems on circulant graphs. Linear Algebra Appl., 285(1-3): 123–142, 1998) showed that computing the clique number and chromatic number are still NP-Hard probl...

Let G be a simple graph with vertex set {v1, v2, … , vn}. The common neighborhood graph of G, denoted by con(G), is a graph with vertex set {v1, v2, … , vn}, in which two vertices are adjacent if and only if they have at least one common neighbor in the graph G. In this paper, we compute the common neighborhood of some composite graphs. In continue, we investigate the relation between hamiltoni...

If a graph has no induced subgraph isomorphic to any graph in a finite family {H1, . . . ,Hp}, it is said to be (H1, . . . ,Hp)-free. The class of H-free graphs has bounded clique-width if and only if H is an induced subgraph of the 4-vertex path P4. We study the (un)boundedness of the clique-width of graph classes defined by two forbidden induced subgraphs H1 and H2. Prior to our study it was ...

We de*ne the clique-width of a countable graph. We prove that a countable graph has *nite clique-width i+ its *nite induced subgraphs have bounded clique-width. We obtain an application to a conjecture concerning the structure of sets of countable graphs having a decidable monadic second-order satis*ability problem. c © 2003 Elsevier B.V. All rights reserved.

‎the narumi-katayama index was the first topological index defined‎ ‎by the product of some graph theoretical quantities‎. ‎let $g$ be a ‎simple graph with vertex set $v = {v_1,ldots‎, ‎v_n }$ and $d(v)$ be‎ ‎the degree of vertex $v$ in the graph $g$‎. ‎the narumi-katayama ‎index is defined as $nk(g) = prod_{vin v}d(v)$‎. ‎in this paper,‎ ‎the narumi-katayama index is generalized using a $n$-ve...

This study examined whether clique isolation predicted an increase in depressive symptoms and whether this association was mediated by loneliness and perceived social acceptance in 310 children followed from age 11-14 years. Clique isolation was identified through social network analysis, whereas depressive symptoms, loneliness, and perceived social acceptance were assessed using self ratings. ...

A graph is clique-perfect if the cardinality of a maximum clique-independent set equals the cardinality of a minimum clique-transversal, for all its induced subgraphs. A graph G is coordinated if the chromatic number of the clique graph of H equals the maximum number of cliques of H with a common vertex, for every induced subgraph H of G. Coordinated graphs are a subclass of perfect graphs. The...