Kautz graph
The Kautz graph K M N + 1 {\displaystyle K_{M}^{N+1}} is a directed graph of degree M {\displaystyle M} and dimension N + 1 {\displaystyle N+1} , which has ( M + 1 ) M N {\displaystyle (M+1)M^{N}} vertices labeled by all possible strings s 0 ⋯ s N {\displaystyle s_{0}\cdots s_{N}} of length N + 1 {\displaystyle N+1} which are composed of characters s i {\displaystyle s_{i}} chosen from an alphabet A {\displaystyle A} containing M + 1 {\displaystyle M+1} distinct symbols, subject to the condition that adjacent characters in the string cannot be equal ( s i ≠ s i + 1 {\displaystyle s_{i}\neq s_{i+1}} ). The Kautz graph K M N + 1 {\displaystyle K_{M}^{N+1}} has ( M + 1 ) M N + 1 {\displaystyle (M+1)M^{N+1}} edges { ( s 0 s 1 ⋯ s N , s 1 s 2 ⋯ s N s N + 1 ) | s i ∈ A s i ≠ s i + 1 } {\displaystyle \{(s_{0}s_{1}\cdots s_{N},s_{1}s_{2}\cdots s_{N}s_{N+1})|\;s_{i}\in A\;s_{i}\neq s_{i+1}\}\,} It is natural to label each such edge of K M N + 1 {\displaystyle K_{M}^{N+1}} as s 0 s 1 ⋯ s N + 1 {\displaystyle s_{0}s_{1}\cdots s_{N+1}} , giving a one-to-one correspondence between edges of the Kautz graph K M N + 1 {\displaystyle K_{M}^{N+1}} and vertices of the Kautz graph K M N + 2 {\displaystyle K_{M}^{N+2}} .