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A well‐known theorem of Whitney states that a 3‐connected planar graph admits an essentially unique embedding into the 2‐sphere. We prove a 3‐dimensional analogue: a simply‐connected 2‐complex every link graph of which is 3‐connected admits an essentially unique locally flat embedding into the 3‐sphere, if it admits one at all. This can be thought of as a generalisation of the 3‐dimensional Schoenflies theorem. [ABSTRACT FROM AUTHOR]

We introduce a strengthening of the notion of transience for planar maps in order to relax the standard condition of bounded degree appearing in various results, in particular, the existence of Dirichlet harmonic function s proved by Benjamini and Schramm. As a corollary we obtain that every planar nonamenable graph admits nonconstant Dirichlet harmonic function s27. [ABSTRACT FROM AUTHOR]

We exhibit a close connection between hitting times of the simple random walk on a graph, the Wiener index, and related graph invariants. In the case of trees, we obtain a simple identity relating hitting times to the Wiener index. It is well known that the vertices of any graph can be put in a linear preorder so that vertices appearing earlier in the preorder are 'easier to reach' by a random walk, but 'more difficult to get out of.' We define various other natural preorders and study their relationships. These preorders coincide when the graph is a tree, but not necessarily otherwise. Our treatise is self-contained, and puts some known results relating the behavior or random walk on a graph to its eigenvalues in a new perspective. [ABSTRACT FROM AUTHOR]

We show that if the sum of the resistances of an electrical network N is finite, then there is a unique electrical current in N, provided that we do not allow, in a sense made precise in the paper, any flow to escape to infinity. [ABSTRACT FROM PUBLISHER]