Counting geodesic paths in graphs

Abstract

A geodesic is a shortest path which connects a pair of vertices of a graph G. In this paper we define the geodesic subpath number gpn(G) of a graph G as the number of geodesics in G. The number of subtrees and subpaths are already studied in literature, but they are both large quantities. Hence, the geodesic subpath number which is related to these quantities but smaller than both, seems worthy of investigation. We first consider extremal graphs with respect to the geodesic subpath number among all connected graphs on n vertices. This number is minimized by the so called geodetic graphs, i.e. graphs in which each pair of vertices is connected by precisely one geodesic. As for the graphs which maximize the geodesic subpath number, we provide an upper bound on gpn(G) in terms of n and we further consider several graph families which might have a large gpn(G). Yet, their value of gpn(G) still does not attain the established bound, so narrowing the gap remains as an open problem. We also consider the class of cactus graphs on n vertices and k cycles and among them characterize extremal graphs with respect to this new invariant.

Figures (5)

• Figure 1: The figure shows two cactus graphs and the unipath structures of these graphs are shaded. Both unipath structures of graph in b) contain only one squared vertex, but it belongs to more than one sqare. The graph a) contains a unipath structure with more than one squared vertex. None of these unipath structures is good.
• Figure 2: The figure shows a unipath resolved cactus graph. It contains two unipath structures and both of them are good.
• Figure 3: The figure shows a squared chain.
• Figure 4: The figure shows all maximal cacti for $n=17$ and $k=7.$
• Figure 5: The figure shows all maximal cacti for $n=9$ and $k=2.$

Theorems & Definitions (11)

• Theorem 2
• Proposition 3
• Proposition 4
• Proposition 5
• Lemma 7
• Lemma 8
• Lemma 9
• Lemma 10
• Lemma 11
• Lemma 12