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This two-graph is a regular two-graph since each pair of distinct vertices appears together in exactly two triples. Given a simple graph G = (V,E), the set of triples of the vertex set V whose induced subgraph has an odd number of edges forms a two-graph on the set V. Every two-graph can be represented in this way. [1]
In general, a subdivision of a graph G (sometimes known as an expansion [2]) is a graph resulting from the subdivision of edges in G. The subdivision of some edge e with endpoints {u,v } yields a graph containing one new vertex w, and with an edge set replacing e by two new edges, {u,w } and {w,v }. For directed edges, this operation shall ...
1. A graph power G k of a graph G is another graph on the same vertex set; two vertices are adjacent in G k when they are at distance at most k in G. A leaf power is a closely related concept, derived from a power of a tree by taking the subgraph induced by the tree's leaves. 2.
Two graphs G and H are homomorphically equivalent if G → H and H → G. [4] The maps are not necessarily surjective nor injective. For instance, the complete bipartite graphs K 2,2 and K 3,3 are homomorphically equivalent: each map can be defined as taking the left (resp. right) half of the domain graph and mapping to just one vertex in the left (resp. right) half of the image graph.
The "similarity" between two graphs should be independent of the total number of nodes or edges, and should depend only upon the differences between relative frequencies of graphlets. Thus, relative graphlet frequency distance D(G,H) between two graphs G and H is defined as:
A Cartesian product of two graphs. In graph theory, the Cartesian product G H of graphs G and H is a graph such that: the vertex set of G H is the Cartesian product V(G) × V(H); and; two vertices (u,v) and (u' ,v' ) are adjacent in G H if and only if either u = u' and v is adjacent to v' in H, or; v = v' and u is adjacent to u' in G.
A graph invariant is additive if, for all two graphs G and H, the value of the invariant on the disjoint union of G and H is the sum of the values on G and on H. For instance, the number of vertices is additive. [1] A graph invariant is multiplicative if, for all two graphs G and H, the value of the invariant on the disjoint union of G and H is ...
The closely related problem of counting the number of isomorphic copies of a graph H in a larger graph G has been applied to pattern discovery in databases, [8] the bioinformatics of protein-protein interaction networks, [9] and in exponential random graph methods for mathematically modeling social networks. [10]