Ring index of a graph

Abstract

Let G be a graph with n vertices and m edges and r connected components. The free rank of G, denoted by $\text {frank}(G)$, is the number of primitive cycles of G. Also, the cycle rank of G was defined as $\text {rank}(G) := m-n+r$. The family of graphs satisfying the equality $\text {rank}(G) = \text {frank}(G)$ is called ring graphs. The full characterization of this family of graphs were given in Gitler et al. (J. Algebraic Comb. 38:721–744, 2013; Discret. Math. 310:430–441, 2010). In this paper, we consider the following problem: Find the minimum number n such that the n-iterated line graph of G is not a ring graph, i.e. $\text {rank}(L^n(G)) \ne \text {frank}(L^n(G))$. For this purpose, we define the ring index of G as the minimum number n such that the n-iterated line graph of G is not a ring graph. We show that the ring index of a graph is at most 3 or is $\infty$. Furthermore, we give a full characterization of graphs with respect to this index.