Combinatorial Properties for a Class of Simplicial Complexes Extended from Pseudo-fractal Scale-free Web

TL;DR

This paper investigates combinatorial properties of a class of simplicial complexes derived from an extended pseudo-fractal scale-free web, providing explicit formulas for various graph invariants and structural counts.

Contribution

It introduces a novel class of simplicial complexes based on a graph product extension and derives explicit combinatorial formulas for key properties.

Findings

Explicit formulas for independence, domination, and chromatic numbers.

Closed-form expressions for acyclic orientations and spanning trees.

Analysis of perfect matchings in specific cases.

Abstract

Simplicial complexes are a popular tool used to model higher-order interactions between elements of complex social and biological systems. In this paper, we study some combinatorial aspects of a class of simplicial complexes created by a graph product, which is an extension of the pseudo-fractal scale-free web. We determine explicitly the independence number, the domination number, and the chromatic number. Moreover, we derive closed-form expressions for the number of acyclic orientations, the number of root-connected acyclic orientations, the number of spanning trees, as well as the number of perfect matchings for some particular cases.