Gallai simplicial complexes

Article type: Research Article

Authors: Muhmood, Shahid | Ahmed, Imran^{; *} | Liaquat, Adnan

Affiliations: Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Defence Road, Off Raiwind Road, Lahore, Pakistan

Correspondence: [*] Corresponding author. Imran Ahmed, Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Defence Road, Off Raiwind Road, Lahore, Pakistan. E-mail: [email protected].

Abstract: Let ΔΓ (G) be Gallai simplicial complex defined as a generalization of Gallai graph Γ (G) such that G is a finite simple graph. The Betti numbers are topological objects which were proved to be invariants by Poincaré. In this note, we give formula for Betti numbers of Gallai simplicial complex associated to prism graph by coding in MATLAB. We prove that a simple graph G having n vertices of the form n = 3l + 2 or 3l + 3 is f-Gallai graph for G=𝕊4l (respectively G=𝕊4l+1 ) a graph consisting of star graphs S2l and S2l′ (respectively S2l and S2l+1) with l common vertices such that l ≥ 2.

Keywords: f-vector, Betti number, Gallai simplicial complex, f-graph

DOI: 10.3233/JIFS-181478

Journal: Journal of Intelligent & Fuzzy Systems, vol. 36, no. 6, pp. 5645-5651, 2019