On a new cyclicity measure of graphs—The global cyclicity index

Being motivated in terms of mathematical concepts from the theory of electrical networks, Klein and Ivanciuc introduced and studied a new graph-theoretic cyclicity index—the global cyclicity index (Klein, Ivanciuc, 2001). 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 Nordhaus–Gaddum-type results. In particular, we establish a relationship between the global cyclicity index C(G)C(G) and the cyclomatic number μ(G)μ(G) of a connected graph GG with nn vertices and mm edges: mn−1μ(G)≤C(G)≤n2μ(G).