Editing Strongly regular graph (section)

===Adjacency matrix equations===
Let ''I'' denote the [[identity matrix]] and let ''J'' denote the [[matrix of ones]], both matrices of order ''v''.  The [[adjacency matrix]] ''A'' of a strongly regular graph satisfies two equations.

First:
:<math>AJ = JA = kJ,</math>
which is a restatement of the regularity requirement. This shows that ''k'' is an eigenvalue of the adjacency matrix with the all-ones eigenvector.

Second:
:<math>A^2 = kI + \lambda{A} + \mu(J - I - A)</math>
which expresses strong regularity. The ''ij''-th element of the left hand side gives the number of two-step paths from ''i'' to ''j''. The first term of the right hand side gives the number of two-step paths from ''i'' back to ''i'', namely ''k'' edges out and back in. The second term gives the number of two-step paths when ''i'' and ''j'' are directly connected. The third term gives the corresponding value when ''i'' and ''j'' are not connected. Since the three cases are [[mutually exclusive]] and [[collectively exhaustive]], the simple additive equality follows.

Conversely, a graph whose adjacency matrix satisfies both of the above conditions and which is not a complete or null graph is a strongly regular graph.<ref>{{citation|first1=P.J.|last1=Cameron|first2=J.H.|last2=van Lint|title=Designs, Graphs, Codes and their Links|publisher=Cambridge University Press|series=London Mathematical Society Student Texts 22|year=1991|isbn=978-0-521-42385-4|page=[https://archive.org/details/designsgraphscod0000came/page/37 37]|url=https://archive.org/details/designsgraphscod0000came/page/37}}</ref>