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Key facts

Vertices

n

Edges

n ( n − 1 ) 2 {\displaystyle \textstyle {\frac {n(n-1)}{2}}}

Radius

{ 0 n ≤ 1 1 otherwise {\displaystyle \left\{{\begin{array}{ll}0&n\leq 1\\1&{\text{otherwise}}\end{array}}\right.}

Diameter

{ 0 n ≤ 1 1 otherwise {\displaystyle \left\{{\begin{array}{ll}0&n\leq 1\\1&{\text{otherwise}}\end{array}}\right.}

Girth

{ ∞ n ≤ 2 3 otherwise {\displaystyle \left\{{\begin{array}{ll}\infty &n\leq 2\\3&{\text{otherwise}}\end{array}}\right.}

Automorphisms

n ! ( S n )

Chromatic number

n

Chromatic index

n if n is odd n − 1 if n is even

Spectrum

{ ∅ n = 0 { 0 1 } n = 1 { ( n − 1 ) 1 , − 1 n − 1 } otherwise {\displaystyle \left\{{\begin{array}{lll}\emptyset &n=0\\\left\{0^{1}\right\}&n=1\\\left\{(n-1)^{1},-1^{n-1}\right\}&{\text{otherwise}}\end{array}}\right.}

Properties

( n − 1) -regular Symmetric graph Vertex-transitive Edge-transitive Strongly regular Integral