Discrete Green's Functions

We study discrete Green's functions and their relationship with discrete Laplace equations. Several methods for deriving Green's functions are discussed. Green's functions can be used to deal with diffusion-type problems on graphs, such as chip-firing, load balancing, and discrete Markov chains.

[1]  Thomas Archibald.  Connectivity and Smoke-Rings: Green's Second Identity in Its First Fifty Years , 1989 .

[2]  Dale H. Mugler,et al.  Green’s Functions for the Finite Difference Heat, Laplace and Wave Equations , 1984 .

[3]  F. Chung,et al.  A Harnack inequality for homogeneous graphs and subgraphs , 1994 .

[4]  G. T. McAllister,et al.  Discrete Green’s functions , 1973 .

[5]  Charles R. Deeter,et al.  The discrete Green's function and the discrete kernel function , 1974, Discret. Math..

[6]  Wolf-Jürgen Beyn.  Discrete Green's functions and strong stability properties of the finite difference method , 1982 .

[7]  H. McKean,et al.  Curvature and the Eigenvalues of the Laplacian , 1967 .

[8]  Jeremy Gray.  Green and Green's functions , 1994 .

[9]  Gordon F. Royle,et al.  Algebraic Graph Theory , 2001, Graduate texts in mathematics.

[10]  William J. Thoms,et al.  George a green , 1857 .

[11]  G. Kirchhoff.  Ueber die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Ströme geführt wird , 1847 .

[12]  F. Chung,et al.  Eigenvalue inequalities for graphs and convex subgraphs , 1997 .

[13]  R. Kress,et al.  Convergence of Discrete Green's Functions for Finite Difference Schemes , 1985 .

[14]  Fan Chung Graham,et al.  A Combinatorial Laplacian with Vertex Weights , 1996, J. Comb. Theory, Ser. A.

[15]  B. McKay,et al.  Feasibility conditions for the existence of walk-regular graphs , 1980 .

[16]  W. O. Storer,et al.  George Green, mathematician and physicist 1793 – 1841 , 1993, The Mathematical Gazette.

[17]  Shing-Tung Yau,et al.  Coverings, Heat Kernels and Spanning Trees , 1998, Electron. J. Comb..

[18]  F. Chung.  A combinatorial trace formula ∗ , 2022 .

[19]  H. Weyl,et al.  Das asymptotische verteilungsgesetz der eigenschwingungen eines beliebig gestalteten elastischen körpers , 1915 .

[20]  Shirley Dex,et al.  JR 旅客販売総合システム（マルス）における運用及び管理について , 1991 .