the inflation $g_{i}$ of a graph $g$ with $n(g)$ vertices and $m(g)$ edges is obtained from $g$ by replacing every vertex of degree $d$ of $g$ by a clique, which is isomorph to the complete graph $k_{d}$, and each edge $(x_{i},x_{j})$ of $g$ is replaced by an edge $(u,v)$ in such a way that $uin x_{i}$, $vin x_{j}$, and two different edges of $g$ are replaced by non-adjacent edges of $g_{i}$. t...

Bayesian networks (BNs) are used to represent and efficiently compute with multi-variate probability distributions in a wide range of disciplines. One of the main approaches to perform computation in BNs is clique tree clustering and propagation. In this approach, BN computation consists of propagation in a clique tree compiled from a Bayesian network. There is a lack of understanding of how cl...

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 maximum edge clique partitioning problem (Max-ECP) was considered by Dessmark et al. (2007) as a tool for DNA clone classification, and is known to be NP-hard on graphs with clique number 3. This complexity result was demonstrated by Punnen and Zhang (2012), however, their proof is incorrect. In this note, we give several complexity results of Max-ECP with respect to clique number, which in...

Graph theory can give a representation of abstract mathematical systems such as groups or rings. We have many graph representations for group, in this study we use the coprime generalized quaternion group to find numerical invariants graph, which are clique number and chromatic number. The main results obtained from is equal each case order.

Given a simple, undirected graph G, Budinich [Discrete Applied Mathematics, 127 (2003), 535 – 543] proposed a lower bound on the clique number of G by combining the quadratic programming formulation of the clique number due to Motzkin and Straus (1965) with the spectral decomposition of the adjacency matrix of G. This lower bound improves the previously known spectral lower bounds on the clique...

Polygon-circle graphs are intersection graphs of polygons inscribed in a circle. This class of graphs includes circle graphs (intersection graphs of chords of a circle), circular arc graphs (intersection graphs of arcs on a circle), chordal graphs and outerplanar graphs. We investigate binding functions for chromatic number and clique covering number of polygon-circle graphs in terms of their c...

The sphericity sph(G) of a graph G is the minimum dimension d for which G is the intersection graph of a family of congruent spheres in Rd . The edge clique cover number (G) is the minimum cardinality of a set of cliques (complete subgraphs) that covers all edges of G. We prove that if G has at least one edge, then sph(G) (G). Our upper bound remains valid for intersection graphs defined by bal...

A graph is called H-free if it contains no copy of H. Denote by fn(H) the number of (labeled) H-free graphs on n vertices. Erdős conjectured that fn(H) ≤ 2(1+o(1)) ex(n,H). This was first shown to be true for cliques; then, Erdős, Frankl, and Rödl proved it for all graphs H with χ(H) ≥ 3. For most bipartite H, the question is still wide open, and even the correct order of magnitude of log2 fn(H...