On the asymptotic behavior of the average geodesic distance L and the compactness C(B) of simple connected undirected graphs whose order approaches infinity

The average geodesic distance L Newman (2003) and the compactness C(B) Botafogo (1992) are important graph indices in applications of complex network theory to real-world problems. Here, for simple connected undirected graphs G of order n, we study the behavior of L(G) and C(B)(G), subject to the condition that their order |V(G)| approaches infinity. We prove that the limit of L(G)/n and C(B)(G) lies within the interval [0;1/3] and [2/3;1], respectively. Moreover, for any not necessarily rational number β ∈ [0;1/3] (α ∈ [2/3;1]) we show how to construct the sequence of graphs {G}, |V(G)| = n → ∞, for which the limit of L(G)/n (C(B)(G)) is exactly β (α) (Theorems 1 and 2). Based on these results, our work points to novel classification possibilities of graphs at the node level as well as to the information-theoretic classification of the structural complexity of graph indices.