The number of walks in a graph Original Research Article

The aim of this note is to call attention to a simple regularity regarding the number of walks in a finite graph G. Let Wk denote the number of walks of length k(≥ 0) in G. Then Wa+b2 ≤ W2aW2b holds for all a, b ϵ View the MathML source0 while equality holds exclusively either 1. (I) for all a, b ϵ View the MathML source0 (in case G is a regular graph), or 2. (II) for all a, b ϵ View the MathML source, or 3. (III) for all a, b ϵ View the MathML source0 of equal parity (provided G is connected, this holds if and only if G is nonregular, yet semiregular graph), or 4. (IV) for all a, b ϵ View the MathML source of equal parity, or 5. (V) just for a = b only. We show that all of these five cases can actually occur and discuss the resulting classification graphs in exactly five classes.