An oriented graph is said positively multiplicative when its adjacency matrix $A$ embeds in a matrix algebra admitting a basis $\mathsf {b}$ with nonnegative structure constants in which the matrix of the multiplication by $A$ coincides with $A$. The goal of this paper is to present basic properties of this notion and explain, through various simple examples, how it relates to highly non trivial problems like the combinatorial description of fusion rules, the description of the minimal boundary of graded graphs or the study of random walks on alcove tilings.
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Keywords: Associative algebras, graphs, harmonic functions, fusion rules
Jérémie Guilhot 1 ; Cédric Lecouvey 1 ; Pierre Tarrago 2
CC-BY-NC-ND 4.0
Jérémie Guilhot; Cédric Lecouvey; Pierre Tarrago. Basics on positively multiplicative graphs and algebras. Confluentes Mathematici, Tome 18 (2026), pp. 31-69. doi: 10.5802/cml.104
@article{CML_2026__18__31_0,
author = {J\'er\'emie Guilhot and C\'edric Lecouvey and Pierre Tarrago},
title = {Basics on positively multiplicative graphs and algebras},
journal = {Confluentes Mathematici},
pages = {31--69},
year = {2026},
publisher = {Institut Camille Jordan},
volume = {18},
doi = {10.5802/cml.104},
language = {en},
url = {https://cml.centre-mersenne.org/articles/10.5802/cml.104/}
}
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