Paths and animals in unbounded degree graphs with repulsion

TL;DR

This paper studies infinite graphs with unbounded degrees where high-degree vertices repel each other, proving exponential bounds on the number of certain subgraphs and paths, with applications to graph indices and growth estimates.

Contribution

It introduces a new class of graphs with degree-based vertex repulsion and establishes exponential bounds on animals and paths, extending understanding of their combinatorial properties.

Findings

Number of animals of size N is exponentially bounded under strong repulsion conditions.

Number of simple paths from a vertex is exponentially bounded under milder conditions.

Results apply to estimates of Randić index growth and greedy animals.

Abstract

A class of countable infinite graphs with unbounded vertex degree is considered. In these graphs, the vertices of large degree `repel' each other, which means that the path distance between two such vertices cannot be smaller than a certain function of their degrees. Assuming that this function increases sufficiently fast, we prove that the number of finite connected subgraphs (animals) of order N containing a given vertex x is exponentially bounded in N for N belonging to an infinite subset N_x of natural numbers. Under a less restrictive condition, the same result is obtained for the number of simple paths originated at a given vertex. These results are then applied to a number of problems, including estimating the growth of the Randi\'c index and of the number of greedy animals.