2021/03/26 by Hibi, Takayuki, Madani, Sara Saeedi
#Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics #Primary 05E40 #Secondary 05C75
paper · doi:10.48550/arxiv.2103.14260
Let G be a finite simple non-complete connected graph on \1, …, n\ and κ(G) ≥ 1 its vertex connectivity. Let f(G) denote the number of free vertices of G and diam(G) the diameter of G. Being motivated by the computation of the depth of the binomial edge ideal of G, the possible sequences (n, q, f, d) of integers for which there is a finite simple non-complete connected graph G on \1, …, n\ with q = κ(G), f = f(G), d = diam(G) satisfying f + d = n + 2 - q will be determined. Furthermore, finite simple non-complete connected graphs G on \1, …, n\ satisfying f(G) + diam(G) = n + 2 - κ(G) will be classified.