Recently, I am interested in the polynomial polynomial of the product of cycles. Let $G = (V , E)$ be an undirected multi-graph with vertex set $\{1,\cdots,n\}$. The graph polynomial of $G$ is defined by $$f_G(x_1,x_2,\cdots,x_n)=\prod_{1\leq i<j\leq n, (i,j)\in E}(x_i-x_j).$$

Conjecture: Let $G $ be the Cartesian product graph $C_{2n+1}\Box C_{2m}$, then the coefficient of $x_1^2x_2^2\cdots x_{(2n+1)(2m)}^2$ in the the graph polynomial $f_G(x_1,x_2,\cdots, x_{(2n+1)(2m)})$ is nonzero.

For $C_3\Box C_{2n}$, the conjecture is true. See the coefficient of a special term in the expansion of the graph polynomial

This conjecture generalized the result about $C_3\Box C_{2n}$. I think it may be true. But I have no idea about the proof on the general cases.

I hope someone could give suggestions about the conjecture. I will appreciate it even if given some special cases for the conjecture such as $n=2,3$,etc.


Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.