# The maximum number of cliques in dense graphs

```@article{Hedman1985TheMN,
title={The maximum number of cliques in dense graphs},
author={Bruce Hedman},
journal={Discret. Math.},
year={1985},
volume={54},
pages={161-166}
}```
• Bruce Hedman
• Published 1985
• Mathematics, Computer Science
• Discret. Math.
Abstract Denote the number of vertices of G by | G |. A clique of graph G is a maximal complete subgraph. The density ω ( G ) is the number of vertices in the largest clique of G . If ω(G)⩾ 1 2 |G| , then G has at most 2 | G |− ω ( G ) cliques. The extremal graphs are then examined as well.
27 Citations

#### Topics from this paper

Clique graphs of packed graphs
• I. Sato
• Computer Science, Mathematics
• Discret. Math.
• 1986
This work corrects the characterization of clique graphs of packed graphs given in Theorem 3.2 of Hedman [3]. Expand
Enumeration of packed graphs
• I. Sato
• Computer Science, Mathematics
• Discret. Math.
• 1990
The number of non-isomorphic (p, n)-packed graphs is obtained by solving the inequality of the following type: For α ≥ 1, β ≥ 1 using LaSalle's inequality. Expand
Another extremal problem for Turan graphs
Abstract We consider only finite, undirected graphs without loops or multiple edges. A clique of a graph G is a maximal complete subgraph of G . The clique number w ( G ) is the number of vertices inExpand
On the Maximum Number of Cliques in a Graph
• D. Wood
• Mathematics, Computer Science
• Graphs Comb.
• 2007
The maximum number of cliques in a graph for the following graph classes is determined: graphs with n vertices and m edges, d-degenerate graphs, and planar graphs. Expand
Integers for the Number of Maximal Independent Sets in Graphs
Let G be a simple undirected graph. Denote by mi(G) (respectively, xi(G)) the number of maximal (respectively, maximum) independent sets in G. In this paper we determine the third and fouth largestExpand
On the number of maximum independent sets of graphs
• Mathematics
• 2014
Let \$G\$ be a simple graph‎. ‎An independent set is a set of‎ ‎pairwise non-adjacent vertices‎. ‎The number of vertices in a maximum independent set of \$G\$ is‎ ‎denoted by \$alpha(G)\$‎. ‎In thisExpand
An upper bound on the number of cliques in a graph
• Mathematics, Computer Science
• Networks
• 1993
It is proved that if the complement of a graph G on n vertices contains no set of t + 1 pairwise disjoint edges as an induced subgraph, then G has fewer than (n/2t)2t maximal complete subgraphs. Expand
THE NUMBER OF MAXIMUM INDEPENDENT SETS IN GRAPHS
• Mathematics
• 2000
In this paper, we study the problem of determining the largest number of maximum independent sets of a graph of order n. Solutions to this problem are given for various classes of graphs, includingExpand
Constraints on the number of maximal independent sets in graphs
• Jiuqiang Liu
• Mathematics, Computer Science
• J. Graph Theory
• 1994
This work proves two conjectures, suggested by P. Erdos, that the maximum number of maximal independent sets among all graphs of order n in a family Φ is o(3n/3) ifΦ is either a family of connected graphs such that the largest value of maximum degrees among all graph n in Φ are o(n) or a family that approaches infinity as n → ∞. Expand
THE SECOND LARGEST NUMBER OF MAXIMAL INDEPENDENT SETS IN GRAPHS WITH AT MOST k CYCLES
• Mathematics
• 2009
Let \$G\$ be a simple undirected graph. Denote by \$\mbox{ mi}(G)\$ (respectively, \$\mbox{xi}(G)\$) the number of maximal (respectively, maximum) independent sets in \$G\$. In this paper we determine theExpand

#### References

SHOWING 1-9 OF 9 REFERENCES
On cliques in graphs
• Mathematics
• 1965
A clique is a maximal complete subgraph of a graph. The maximum number of cliques possible in a graph withn nodes is determined. Also, bounds are obtained for the number of different sizes of cliquesExpand
An upper bound on the size of the largest cliques in a graph
An upper bound for the number of vertices existing in a clique of maximum cardinal is produced based in particular on the existence of a maximum cardinal clique that contains no vertex x such that the neighborhood of x is contained in the neighborhoods of another vertex y. Expand
On clique-extremal (p, q)-graphs
• Mathematics, Computer Science
• Networks
• 1974
A standard form for such extremal graphs is developed and these standard forms are shown to have a complementary kind of analogy with respect to maximum and minimum. Expand
The maximum number of q-cliques in a graph with no p-clique
It is shown that, except for the trivial case 1 =< n < q, Turan's graph is the unique graph which attains the maximum @?(n, p, q) for all q such that 1 < q < p. Expand
Determining the Stability Number of a Graph
• V. Chvátal
• Mathematics, Computer Science
• SIAM J. Comput.
• 1977
The resulting system is powerful enough to encompass the algorithms of Tarjan''s type and provide very short proofs on graphs for which the stability number equals the clique-covering number. Expand
On cliques in graphs
Sharp bounds are found on the maximal number of sizes of cliques in a graph onn vertices.
Graph theory