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A strongly regular graph is a regular graph where every adjacent pair of vertices has the same number l of neighbors in common, and every non-adjacent pair of vertices has the same number n of neighbors in common. The smallest graphs that are regular but not strongly regular are the cycle graph and the circulant graph on 6 vertices.
The web graph W n,r is a graph consisting of r concentric copies of the cycle graph C n, with corresponding vertices connected by "spokes". Thus W n,1 is the same graph as C n, and W n,2 is a prism. A web graph has also been defined as a prism graph Y n+1, 3, with the edges of the outer cycle removed. [7] [10]
In the mathematical discipline of graph theory, a graph labeling is the assignment of labels, traditionally represented by integers, to edges and/or vertices of a graph. [ 1 ] Formally, given a graph G = ( V , E ) , a vertex labeling is a function of V to a set of labels; a graph with such a function defined is called a vertex-labeled graph .
In graph theory, a walk-regular graph is a simple graph where the number of closed walks of any length from a vertex to itself does only depend on but not depend on the choice of vertex. Walk-regular graphs can be thought of as a spectral graph theory analogue of vertex-transitive graphs .
Download as PDF; Printable version; In other projects Wikimedia Commons; ... Pages in category "Regular graphs" The following 119 pages are in this category, out of ...
Such a strongly regular graph is denoted by srg(v, k, λ, μ). Its complement graph is also strongly regular: it is an srg(v, v − k − 1, v − 2 − 2k + μ, v − 2k + λ). A strongly regular graph is a distance-regular graph with diameter 2 whenever μ is non-zero. It is a locally linear graph whenever λ = 1.
K n has n(n – 1)/2 edges (a triangular number), and is a regular graph of degree n – 1. All complete graphs are their own maximal cliques. They are maximally connected as the only vertex cut which disconnects the graph is the complete set of vertices. The complement graph of a complete graph is an empty graph.
Since = if and only if the graph is bipartite, we will refer to the graphs that satisfy this alternative definition but not the first definition bipartite Ramanujan graphs. If G {\displaystyle G} is a Ramanujan graph, then G × K 2 {\displaystyle G\times K_{2}} is a bipartite Ramanujan graph, so the existence of Ramanujan graphs is stronger.