On the critical group of the k-partite graph

TL;DR

This paper investigates the structure of the critical group of k-partite graphs, providing algorithms and complete characterizations for k from 2 to 6, extending known results for bipartite graphs.

Contribution

It introduces an algorithm to determine the critical groups of k-partite graphs and fully characterizes these groups for k between 2 and 6, generalizing bipartite graph results.

Findings

Complete characterization of critical groups for k=2 to 6

Algorithm for computing Smith normal forms of graph Laplacians

Extension of bipartite graph results to k-partite graphs

Abstract

The critical group of a connected graph is closely related to the graph Laplacian, and is of high research value in combinatorics, algebraic geometry, statistical physics, and several other areas of mathematics. In this paper, we study the k-partite graphs and introduce an algorithm to get the structure of their critical groups by calculating the Smith normal forms of their graph Laplacians. When k is from 2 to 6, we characterize the structure of the critical groups completely, which can generalize the results of the complete bipartite graphs.