Spanning Simplicial Complexes of Uni-Cyclic Multigraphs

TL;DR

This paper introduces spanning simplicial complexes for multigraphs, characterizes spanning trees in uni-cyclic multigraphs, and derives algebraic and topological invariants like facet ideals and Euler characteristics.

Contribution

It generalizes spanning simplicial complexes to multigraphs, characterizes all spanning trees in uni-cyclic multigraphs, and computes algebraic and topological invariants.

Findings

Characterization of all spanning trees in uni-cyclic multigraphs.

Determination of the facet ideal and its primary decomposition.

Formula for the Euler characteristic of the spanning simplicial complex.

Abstract

A multigraph is a nonsimple graph which is permitted to have multiple edges, that is, edges that have the same end nodes. We introduce the concept of spanning simplicial complexes $\Delta_s(\mathcal{G})$ of multigraphs $\mathcal{G}$, which provides a generalization of spanning simplicial complexes of associated simple graphs. We give first the characterization of all spanning trees of a uni-cyclic multigraph $\mathcal{U}_{n,m}^r$ with $n$ edges including $r$ multiple edges within and outside the cycle of length $m$. Then, we determine the facet ideal $I_\mathcal{F}(\Delta_s(\mathcal{U}_{n,m}^r))$ of spanning simplicial complex $\Delta_s(\mathcal{U}_{n,m}^r)$ and its primary decomposition. The Euler characteristic is a well-known topological and homotopic invariant to classify surfaces. Finally, we device a formula for Euler characteristic of spanning simplicial complex $\Delta_s(\mathcal{U}_{n,m}^r)$.