Combinatorial PDEs on Cayley and coset graphs (Q409383)

The paper under review extends the technique of discrete Fourier transforms on \({\mathbb Z}_2^n\) to Cayley and coset graphs. Discrete Fourier and Radon transforms on \({\mathbb Z}^n_2\) were introduced for the needs of code theory [\textit{P. Diaconis} and \textit{R. Graham}, ``The Radon transform on \({\mathbb Z}^N_2\)'', Pac. J. Math. 118, 323--345 (1985; Zbl 0581.43001)] and found successively spectacular applications in graph reconstruction theory [\textit{R. P. Stanley}, ``Reconstruction from vertex switching'', J. Comb. Theory, Ser. B 38, 132-138 (1985; Zbl 0572.05046)]. Starting from the idea that Fourier transforms where originally devised for the purpose of transforming PDEs (on infinite domains) into ODEs, Fourier transforms on \({\mathbb Z}^n_2\) where recognized [\textit{E. Barletta} and \textit{S. Dragomir}, ``Combinatorial PDEs on Hamming graphs'', Discrete Math. 254, No. 1-3, 1--18 (2002; Zbl 1001.05116)] as the appropriate tool for solving discrete versions (as due for instance to \textit{H. Urakawa} [``Heat kernel and Green kernel comparison theorems for infinite graphs'', J. Funct. Anal. 146, No. 1, 206--235 (1997; Zbl 0870.05070)]) of the wave and heat equations on a Hamming graph, with the successive discussion of discrete heat equation morphisms [\textit{V. Abatangelo} and \textit{S. Dragomir}, ``Discrete heat equation morphisms'', Interdiscip. Inf. Sci. 14, No. 2, 225--244 (2008; Zbl 1166.05310)] and again with applications (of discrete Fourier calculus on \({\mathbb Z}_2^n\), rather than the very solutions to discrete PDEs) to the graph reconstruction problem [\textit{V. Abatangelo} and \textit{S. Dragomir}, ``Discrete Fourier calculus and graph reconstruction'', Interdiscip. Inf. Sci. 13, No. 2, 163--180 (2007; Zbl 1144.43002)]. It should be mentioned that discrete heat equation morphisms are related (via the work by \textit{M. Kanai} [``Rough isometries and combinatorial approximations of geometries of non-compact Riemannian manifolds'', J. Math. Soc. Japan 37, 391--413 (1985; Zbl 0554.53030)]) to the work by \textit{E. Loubeau} [``Morphisms of the heat equation'', Ann. Global Anal. Geom. 15, No. 6, 487--496 (1997; Zbl 0910.58009)] and, as well as their counterpart in Riemannian geometry, give rise to discrete harmonic morphisms as introduced by \textit{H. Urakawa} [``A discrete analogue to the harmonic morphism and Green kernel comparison theorems'', Glasg. Math. J. 42, No. 3, 319--334 (2000; Zbl 1002.05049)]. Although direct applications of the very explicit solutions to discrete PDEs (on Hamming, Cayley, or coset graphs) aren't available as yet, I believe the paper under review contributes to opening a wide space for further investigation in a nice area involving Riemannian geometry, PDEs, and discrete Fourier calculus.