Strongly regular graph

This article defines a property that can be evaluated to true/false for any undirected graph, and is invariant under graph isomorphism. Note that the term "undirected graph" as used here does not allow for loops or parallel edges, so there can be at most one edge between two distinct vertices, the edge is completely described by the vertices it joins, and there can be no edge from a vertex to itself.
View other such properties

Definition

Definition for finite graphs

Suppose $G$ is a finite undirected graph with $n$ vertices. Suppose $d,\lambda ,\mu$ are nonnegative integers. We say that $G$ is a strongly regular graph of type $(n,d,\lambda ,\mu )$ (we sometimes write this as ${\mbox{srg}}(n,d,\lambda ,\mu )$) if it satisfies all of the following conditions:

1. $G$ is a $d$-regular graph, i.e., the degree of every vertex of $G$ equals $G$.
2. Any two adjacent vertices of $G$ have precisely $\lambda$ vertices that are adjacent to both of them.
3. Any two non-adjacent vertices of $G$ have precisely $\mu$ vertices that are adjacent to both of them.

We say that a graph is strongly regular if we can find values of the parameters above for which the graph is strongly regular of that type.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
complete graph
empty graph