2017/02/15 by Ahmed, Imran, Muhmood, Shahid
#05E25 #06A11 #13H10 (Secondary) #13P10 (Primary) #55U10 #Algebraic Topology (math.AT) #Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.1702.04623
Let G be a finite simple graph. The line graph L(G) represents the adjacencies between edges of G. We define first the line simplicial complex ΔL(G) of G containing Gallai and anti-Gallai simplicial complexes ΔΓ(G) and ΔΓ'(G) (respectively) as spanning subcomplexes. The study of connectedness of simplicial complexes is interesting due to various combinatorial and topological aspects. In Theorem 3.3, we prove that the line simplicial complex ΔL(G) is connected if and only if G is connected. In Theorem 3.4, we establish the relation between Euler characteristics of line and Gallai simplicial complexes. In Section 4, we discuss the shellability of line and anti-Gallai simplicial complexes associated to various classes of graphs.