No abstract

A graph G has a finite set V of points and a set X of lines each of which joins two distinct points (called its end-points), and no two lines join the same pair of points. A graph with one point and no line is trivial. A line is incident with each of its end-points. Two points are adjacent if they are joined by a line. The degree of a point is the number of lines incident with it. The line-graph L(G) of G has X as its set of points and two elements x, y of X are adjacent in L(G) whenever the lines x and y of G have a common end-point. A walk in G is an alternating sequence v1, x1, v2, x2, …, vn of points and lines, the first and last terms being points, such that xi is the line joining vi to vi+1 for i=1, …, n-1.

No abstract

Let G be a locally finite connected graph and c be a positive real-valued function defined on its edges. Let D(ξ) denote the sum of the values of c on the edges incident with a vertex ξ. A particle starts at some vertex α and performs an infinite random walkin which (i) the ξj are vertices of G, (ii), λj. is an edge joining ξj–1 to ξj (j = 1, 2, 3, …), (iii) if λ is any edge incident with ξj, thenLet υ be a set of vertices of G such that the complementary set of vertices is finite and includes α. A geometrical characterization is given of the probability (τ, say) that the particle will visit some element of υ without first returning to α. An essentially equivalent problem is obtained by regarding G as an electrical network and c(λ) as the conductance of an edge λ; the current flowing through the network from α to υ when an external agency maintains α at potential I and all elements of υ at potential 0 is found to be τD(α).A necessary and sufficient condition (of a geometrical character) for the particle to be certain to return to α. is obtained; and, as an application, a new proof is given of a conjecture of Gillis (3) regarding centrally biased random walk on an n–dimensional lattice.

A new and simple proof is given of the known theorem that, if T1, T2,… is an infinite sequence of finite trees, then there exist i and j such that i < j and Ti is homeomorphic to a subtree of Tj.