Proof of Kelly-Ulam Conjecture

TL;DR

This paper proves the Kelly-Ulam conjecture by demonstrating that hypomorphic graphs have identical counts of l-paths between vertices, confirming they are isomorphic for finite simple graphs.

Contribution

The paper establishes that hypomorphic graphs have equal l-path counts and proves the Kelly-Ulam conjecture for all finite simple graphs.

Findings

Hypomorphic graphs have the same number of l-paths between any two vertices.

The Kelly-Ulam conjecture holds for all finite simple graphs.

Hypomorphic graphs are isomorphic in the category of finite simple graphs.

Abstract

The deck of a graph $X$, $D(X)$, is defined as the multiset of all vertex-deleted subgraphs of $X$. Two graphs are said to be hypomorphic, if they have the same deck. Kelly-Ulam conjecture states that any two hypomorphic graphs on at least three vertices are isomorphic. In this paper, we first prove that for two finite simple hypomorphic graphs the number of $l$-paths between two arbitrary vertices are equal, where $1 \leq l \leq n - 2$. As a consequence, it is proved that the Kelly-Ulam conjecture is correct over the category of all finite simple graphs.