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Tension polynomials of graphs

✍ Scribed by Martin Kochol

Publisher
John Wiley and Sons
Year
2002
Tongue
English
Weight
104 KB
Volume
40
Category
Article
ISSN
0364-9024

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✦ Synopsis

The tension polynomial F G (k) of a graph G, evaluating the number of nowhere-zero Z k -tensions in G, is the nontrivial divisor of the chromatic polynomial G (k) of G, in that G (k) ¼ k c(G) F G (k), where c(G) denotes the number of components of G. We introduce the integral tension polynomial I G (k), which evaluates the number of nowhere-zero integral tensions in G with absolute values smaller than k. We show that 2 r (G) F G (k) ! I G (k) ! (r (G) þ 1)F G (k), where r (G) ¼ jV(G)j À c(G), and, for every k > 1, F G (k þ 1) ! F G (k) Á k / (k À 1) and I G (k þ 1) ! I G (k) Á k / (k À 1).

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