Face-Counting Identities and Derivatives of Face Polynomials

Abstract

In this paper, we first introduce two incidence matrices related to the vertex-deck and the edge-deck of a given simplicial complex. Then, we obtain two face-counting identities based on these incidence matrices. Using these face-counting identities, we present combinatorial interpretations of the first and the second derivatives of face polynomials of simplicial complexes. We also propose several interesting open questions and conjectures. Finally, we conclude the paper with a discussion about some possible future research works.

Keywords

• Face polynomial
• face-counting identities
• incidence matrices
• derivatives

References

1. Brown, J.I., Discrete Structures and Their Interactions, Chapman and Hall/CRC, 1st edition, 2013.
2. Kelly, P.J., A congruence theorem for trees, Pacific Journal of Mathematics, 7(1957), 961–968.
3. Knill, O., A parametrized Poincare-Hopf theorem and clique cardinalities of graphs, https://arxiv.org/abs/1906.06611, (2019).
4. Li, X., Gutman, I., A unified approach to the first derivatives of graph polynomials, Discrete Applied Mathematics, 58(1995), 293–297.
5. Li, X., Shi, Y., Derivatives and real roots of graph polynomials, https://arxiv.org/abs/1601.01843, (2016).
6. Stanley, R.P., f-vectors and h-vectors of simplicial posets, Journal of Pure and Applied Algebra, 71(1991), 319–331.
7. Steven, K., Isabella, N., Face enumeration on simplicial complexes, https://arxiv.org/abs/1505.06380, (2015).
8. Teimoori, H., Clique polynomials of 2-connected K5-free chordal graphs, Journal of Algebra Combinatorics Discrete Structures and Applications, 8(2021), 25–31.