Mathematics & Physics 2015, 8(1), 28–30 УДК 513.88 Mean Value Theoremfor Harmonic Functions on Cayley Tree Farrukh T. Ishankulov∗ Samarkand State University Universitetsky bulvar, 15 140104,Samarkand Uzbekistan Received 06.09.2014, received in revised form 06.10.2014, accepted 24.11.2014 An analog of the mean value theorem for harmonic functions on Cayley tree is proved in this paper. <...> Keywords: harmonic function, Caylee tree, mean value. case we write l =< x, y >. <...> A function u : V → R is said to be harmonic on Cayley tree if it satisfies the discrete Laplace equation u(x) = 1 k +1 u(y), y∈W1(x) where W1(x) is the set of vertices adjacent to x. <...> To define a discrete ball and sphere [1] let us fix a point x ∈ V . <...> The mean value theorem plays an important role in the classical potential theory. <...> According to this theorem, the value of harmonic function at the center of a sphere is equal to arithmetic average of its values on the surface of this sphere. <...> This theorem can be also formulated for a ball: the value of harmonic function at the center of the ball is equal to arithmetic average of its values in points of this ball. <...> The aim of this paper is to proof an analog of the mean value theorem for harmonic functions on Cayley tree. ∗fandor83@mail.ru Siberian Federal University. <...> All rights reserved c – 28 – (1) tree is an infinite tree in which each vertex has exactly k +1 incident edges (the Cayley tree of order k 1). <...> Let Γk = (V,L) be the Cayley tree of order k 1, where V and L are the vertex set and the edge set, respectively. <...> If x, y ∈ V are the endpoints of an edge l ∈ L then x and y are said to be adjacent. <...> In this Introduction A tree is connected acyclic graph. <...> One special case of such graph is a Cayley tree. <...> A Cayley Farrukh T. Ishankulov Mean Value Theorem for Harmonic Functions on Cayley Tree 1. <...> If u(x) is a harmonic function <...>