Quotient graphs and stochastic matrices

TL;DR

This paper explores the relationship between graphs with the same quotient structures, developing new theoretical insights involving doubly stochastic matrices and extending to weighted graphs, with applications to quantum walks.

Contribution

It introduces a novel theory linking graphs with the same symmetrized quotient to doubly stochastic matrices, extending equitable partition concepts to weighted graphs and quantum walk applications.

Findings

Characterization of graphs with the same combinatorial quotient

Development of theory connecting symmetrized quotients to doubly stochastic matrices

Application to quantum walks and weighted graphs

Abstract

Whenever graphs admit equitable partitions, their quotient graphs highlight the structure evidenced by the partition. It is therefore very natural to ask what can be said about two graphs that have the same quotient according to certain equitable partitions. This question has been connected to the theory of fractional isomorphisms and covers of graphs in well-known results that we briefly survey in this paper. We then depart to develop theory of what happens when the two graphs have the same symmetrized quotient, proving a structural result connecting this with the existence of certain doubly stochastic matrices. We apply this theorem to derive a new characterization of when two graphs have the same combinatorial quotient, and we also study graphs with weighted vertices and the related concept of pseudo-equitable partitions. Our results connect to known old and recent results, and are naturally applicable to study quantum walks.