In particular, given a new edge ((v_i, v_j)), our approach requires searching only the local neighborhood around that edge (to check if a larger clique of size (k+1) is possible).
A clique of a graph is a complete subgraph and the number of vertices in a largest clique of graph G, denoted by (omega (G)), is called the clique number of G. Recall that a graph is said to be planar if it can be drawn in the plane so that its edges intersect only at their ends.
We then consider pairs that are part of a larger clique closer than pairs that are part of a smaller clique, and consider pairs that are part of an m-core closer than pairs that are part of an n-core if m > n.
A maximal clique in G is a clique whose vertices are not a subset of the vertices of a larger clique, and the maximum cliques in G are the largest among all cliques in a graph.
A maximal clique ω(G m ) in a graph G m is a clique of a graph that is not a subgraph of a larger clique in G m possessing these criteria (i.e. bonds, atom types and their connectivity are preserved), hence this subgraph is referred to as a Maximum Common Subgraph (MCS).
If a subgraph G' constitutes a k-clique and G' is no proper subgraph of a larger clique, it is called a maximal clique.
A maximal clique is a clique that cannot be extended by including one more adjacent vertex, i.e. a clique which does not exist exclusively within the vertex set of a larger clique.
Large cliques of orthologs combined with an additive scoring scheme based on phylogenetic distances between plant species provide the mechanism for gene ontology knowledge transfer from orthologous genes in other well annotated plant species to wheat genes.
This problem is NP-complete, and as such, many consider that it is unlikely that an efficient algorithm for finding the largest clique of a graph exists.
When the sample size is smaller than the number of genes, i.e. n < p, and, in addition, n is smaller than the dimension of the larger clique of the graph, a shrinkage covariance matrix with null elements corresponding to the missing edges of the graph G may be employed.
