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**1 - 7**of**7**### Paths and animals in unbounded degree graphs with repulsion

"... A class of countable infinite graphs with unbounded vertex degree is con-sidered. 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 in ..."

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A class of countable infinite graphs with unbounded vertex degree is con-sidered. 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 (an-imals) of order N containing a given vertex x is exponentially bounded in N for N belonging to an infinite subset Nx ⊂ N. Under a less restrictive condi-tion, 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 Randic ́ index and of the number of greedy animals.