# Conditionally monotone independence and the associated products of graphs

## TL;DR

This paper introduces the c-comb (loop) product of birooted graphs, connecting it to c-monotone convolution of distributions, by reducing c-monotone independence to tensor independence.

## Contribution

It generalizes graph products by linking them to c-monotone independence and convolution, using tensor product realization for the first time in this context.

## Key findings

- Introduces the c-comb (loop) product of birooted graphs.
- Establishes the relation between graph products and c-monotone convolution.
- Reduces c-monotone independence to tensor independence.

## Abstract

We reduce the conditionally monotone (c-monotone) independence of Hasebe to tensor independence. For that purpose, we use the approach developed for the reduction of boolean, free and monotone independences to tensor independence. We apply the tensor product realization of c-monotone random variables to introduce the c-comb (loop) product of birooted graphs, a generalization of the comb (loop) product of rooted graphs, and we show that it is related to the c-monotone additive (multiplicative) convolution of distributions.