Further results on the global cyclicity index of graphs

TL;DR

This paper advances the understanding of the global cyclicity index of graphs by establishing new bounds, properties, and relationships with other graph invariants, using techniques from graph theory and electrical network theory.

Contribution

It provides new bounds, properties, and relationships for the global cyclicity index, enhancing the theoretical framework of graph cyclicity measures.

Findings

Established bounds relating cyclicity index and cyclomatic number.

Proved the strictly monotone increasing property of the index.

Derived Nordhaus-Gaddum-type results for the cyclicity index.

Abstract

Being motivated in terms of mathematical concepts from the theory of electrical networks, Klein & Ivanciuc introduced and studied a new graph-theoretic cyclicity index--the global cyclicity index (Graph cyclicity, excess conductance, and resistance deficit, J. Math. Chem. 30 (2001) 271--287). In this paper, by utilizing techniques from graph theory, electrical network theory and real analysis, we obtain some further results on this new cyclicity measure, including the strictly monotone increasing property, some lower and upper bounds, and some Nordhuas-Gaddum-type results. In particular, we establish a relationship between the global cyclicity index $C(G)$ and the cyclomatic number $\mu(G)$ of a connected graph $G$ with $n$ vertices and $m$ edges: $$\frac{m}{n-1}\mu(G)\leq C(G)\leq \frac{n}{2}\mu(G).$$