Graph Theory: Lecture No. 20
Let Gbe a graph with vertex set
V={v1,v2, . . . , vn}. Set X= (x1, . . . , xn). The
adjacency polynomial of Gis the multivariate
polynomial
A(G,X) = Πi<j{(xi−xj) : vivj∈E}
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Graph Theory: Lecture No. 20 - Adjacency Polynomial, Orientations, and List Coloring, Slides of Applied Mathematics
Various topics in graph theory as presented in lecture no. 20. Topics include the adjacency polynomial of a graph, orientations of graphs, weight of sequences, and list coloring. The document also discusses the combinatorial nullstellen szatz and its implications for graph coloring.
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2012/2013
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Let G be a graph with vertex set V = {v 1 , v 2 ,... , vn}. Set X = (x 1 ,... , xn). The adjacency polynomial of G is the multivariate polynomial
A(G , X ) = Πi<j {(xi − xj ) : vi vj ∈ E }
Let D be an orientation of G. Then σ(D) = Π{σ(a) : a ∈ A(D)} where σ(a) = +1 if a = (vi , vj ) with i < j and σ(a) = − 1 if a = (vi , vj ) with i > j.
Setting xd^ = Πni=1xi di
A(G , X ) = Σd w (d)xd
Let f be a nonzero polynomial over a field F in the variables X = (x 1 , x 2 ,... , xn), of degree di in xi , for 1 ≤ i ≤ n. Let Li be a set of di + 1 elements of F , 1 ≤ i ≤ n. Then there exists t ∈ L 1 ×... × Ln such that f (t) 6 = 0.
Suppose G has an orientation D such that its outdegree sequence is d, then:
1 If D′^ is an orientation of G with outdegree sequence d then σ(D′) = σ(D) if and only if |A(D) − A(D′)| is even. 2 If D has no directed odd cycles, then all orientations of G with outdegree sequence d have the same sign.
Let G be a graph and let D be an orientation of G without directed odd cycles. Then G is (d + 1)-list colorable, where d is the outdegree sequence of G.
For any loopless graph G , there exists a polynomial P(G , x) such that P(G , k) = C (G , k) for all non-negative integers k. (Here C (G , k) is the number of distinct proper k-colorings of a graph G ) Moreover, if G is simple, and e is any edge of G , then P(G , x) satisfies the recursion formula:
P(G , x) = P(G − e, x) − P(G /e, x)
The polynomial P(G , x) is of degree n, with integer coefficients which alternate in sign, leading term xn^ and constant term 0.