Inner Local Exponent of Two-coloured Digraphs with Two Cycles of Length n and 4n + 1

A two-coloured digraph D(2) is a digraph in which each arc is coloured with one of two colours – for example, red or black. A two-coloured digraph D(2) is said to be primitive if there are positive integers a and i such that for each pair of points x and y in D(2) there is an (a, i)-walk from x to y. The inner local exponent of a point pv in D(2) denoted by expin(pv,D(2)) is the smallest positive integer a + i over all non-negative integers a and i such that there is a walk from each vertex in D(2) to pv consisting of a red arcs and i black arcs. In a two-coloured primitive digraph, two cycles of length n and 4n+1 result in four or five red arcs. For the two-coloured digraphs, primitivity and inner local exponent are discussed at each point.