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The major points which I found very interactive in learning graph theory are: Direction, Cycle, Traversal, Forward Arc, Reverse Arcs, Circulation, Combination, Digraph, Non-Negative Linear Combination, Non-Negative Circulation
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Let C be a cycle, together with a given sense of traversal. An arc of C is a forward arc if its direction agrees with the sense of traversal of C and a reverse arc otherwise. We denote the set of forward and reverse arcs by C +^ and C − respectively and associate with C the circulation fC defined by fc (a) = 1 if a ∈ C +, fC (a) = − 1 if a ∈ C −^ and fC (a) = 0 otherwise. fC is indeed a circulation.
Every circulation on a digraph is a linear combination of the circulations associated with its cycle.
Let N = N(x, y ) be a network in which each arc is of unit capacity. Then N has an (x, y )-flow of value k if and only if its underlying digraph D(x, y ) has k arc-disjoint
directed (x, y )-paths.
Menger’s Theorem for Directed graphs: In any digraph D(x, y ) the maximum number of
pairwise arc disjoint directed (x, y ) paths is equal to the minimum number of arcs in an (x, y )-cut.
Let g be an element of the row space of M,
i.e. g = pM for some vector p ∈ RV^. If a = (x, y ) is an arc, then g (a) = p(x) − p(y ).
With each bond B one may associate a tension gB defined by: gB (a) = 1 if a is a forward arc, and gB (a) = − 1 if a is a reverse
arc and gB (a) = 0 if a ∈/ B.
Every tension in a digraph is a linear combination of the tensions associated with its bonds. Every non-negative tension in a digraph is a non-negative linear combination of the tensions associated with its directed bonds. Moreover, if the tension is integer valued, the coefficients of the linear combination may be chosen to be non-negative integers.
Let D = (V , A) be a directed graph. Suppose that with each arc a of D, two real numbers b(a) and c(a) are associated such that
b(a) ≤ c(a). A circulation f in D is feasible, with respect to the functions b and c, if b(a) ≤ f (a) ≤ c(a) for all a ∈ A.
Hoffman’s Circulation Theorem: A Digraph D has a feasible circulation with respect to bounds b and c if and only if these bounds satisfy the above inequality. Furthermore if b and c are integer valued and satisfy this inequality, then D has an integer valued feasible circulation
