By Nicos Christofides

It's always useful and visually attractive, to depict a few scenario that's of curiosity by means of a graphical determine along with issues (vertices)—representing entities—and traces (links) becoming a member of definite pairs of those vertices and representing relationships among them. Such figures are recognized through the final identify graphs and this booklet is dedicated to their learn. Graphs are met with in all places below varied names: ''structures'' in civil engineering, ''networks'' in electric engineering, ''sociograms'', ''communication structures'' and ''organizational structures'' in sociology and economics, ''molecular structure'' in chemistry, ''road maps'', gasoline or electrical energy ''distribution networks'' etc.

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Impose a voltage of v(a) at a and 0 on Z. Since v(•) is linear in v(a) by the superposition principle, we have that Px [τa < τZ ] = v(x)/v(a), whence ( ) ∑ c(a, x) [ v(x) ] p(a, x) 1 − Px [τa < τZ ] = 1− π(a) v(a) x x ∑ ∑ 1 1 = c(a, x)[v(a) − v(x)] = i(a, x) . v(a)π(a) x v(a)π(a) x P[a → Z] = ∑ In other words, ∑ v(a) = i(a, x) . 4) c ⃝1997–2014 by Russell Lyons and Yuval Peres. Commercial reproduction prohibited. Version of 27 February 2014. DRAFT §2. More Probabilistic Interpretations 29 where the last notation indicates the dependence on a and Z.

Series Law. Two resistors* r1 and r2 in series are equivalent to a single resistor r1 + r2 . In other words, if w ∈ V(G) \ (A ∪ Z) is a node of degree 2 with neighbors u1 , u2 and we replace the edges (ui , w) by a single edge (u1 , u2 ) having resistance r(u1 , w) + r(w, u2 ), then all potentials and currents in G \ {w} are unchanged and the current that flows from u1 to u2 equals i(u1 , w). u1 w u2 Proof. It suﬃces to check that Ohm’s and Kirchhoﬀ’s laws are satisfied on the new network for the voltages and currents given.

Following the transformations indicated in the figure, we obtain C (a ↔ z) = 7/12, so that P[a → z] = † DRAFT 7/12 C (a ↔ z) 7 = = . π(a) 3 36 A conductor c is an edge with conductance c. c ⃝1997–2014 by Russell Lyons and Yuval Peres. Commercial reproduction prohibited. Version of 27 February 2014. DRAFT 34 Chap. 2: Random Walks and Electric Networks 1 1/2 1 1 1 1 1 1 1 1/2 a z 1 1/2 a 1/2 1/2 1 1 z 1 1 1 1 1/2 1/2 1 1 1 1/4 1 1/2 a 1 a z z 1 1 1/3 a 1 z 7/12 Note that in any network G with voltage applied from a to z, if it happens that v(x) = v(y), then we may identify x and y to a single vertex, obtaining a new network G/{x, y} in which the voltages at all vertices are the same as in G.