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On the properties of the first graph cohomology over the algebra of truncated polynomials A_m
by
Radmila Sazdanovic
George Washington University
Coauthors: Jozef Przytycki, Milena Pabiniak

We created Mathematica package for computing H_A3^{1, (v−1)(m−1)−(m−2)(n−1)/2}(G) and Tor H_Am^{1, (v−1)(m−1)−(m−2)(n−1)/2}(G) for an arbitrary simple graph with v vertices, any n ³ 3 and algebras of truncated polynomials A_m. Obtained results motivated following conjectures:

Conjecture 1. For all simple graphs G and G¹: G Ì G¹ Þ H_A3^{1, 2v(G)−3}(G) Ì H_A3^{1, 2v(G1)−3}(G¹).

We verified this for the complete lattice of all non-isomorphic subgraphs of K₅.

Conjecture 2. For any complete graph with n vertices K_n, n > 3 the following is true: H^{1, 2n−3}_A3(K_n) = Z₃ⁿ⁻¹ ÅZ₂ ÅZ^{n(n−1)(2n−7)/6}.

We verified this for n £ 20 and for K₂₀ indeed we get: H^{1, 37}_A3 = Z₃¹⁹ ÅZ₂ ÅZ²⁰⁹⁰.

Moreover, we will present a few interesting observations:

1. Computations suggest that if K₄ Ì G then Z₂ Ì H_A3^{1, 2v−3} (G). Similarly, if K₅ Ì G then Z₂ ÅZ₃⁴ Ì H_A3^{1, 2v−3}(G).

2. The only example (so far) of Z₄ torsion appears in Tor(H_A5^{1, 14}(K₆)) = Z₅⁴⁰ ÅZ₂²⁶ ÅZ₄.

Date received: December 6, 2005

Copyright © 2005 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # carw-19.