An Intermediate Value Theorem for the Decycling Numbers of Toeplitz Graphs

Abstract

In this paper we obtain an intermediate value theorem for the decycling numbers of Toeplitz graphs: if $$n \ge 3$$n?3, and $$0 \le r \le n - 2$$0≤r≤n-2, then there exists a Toeplitz graph of order $$n$$n with decycling number $$r$$r. We also prove that the decycling numbers of connected Cayley graphs of order $$n$$n satisfy the intermediate value property if and only if $$n = 4$$n=4 or $$6$$6.