2025/01/09 by Bhavale, Ashok Nivrutti · 1 citation
#05C20 #05C78 #06A05 #06A06 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2501.05064
In 1973, Harary and Palmer posed the problem of enumeration of labeled graphs on n ≥ 1 unisolated vertices and l ≥ 0 edges. In 1997, Bender et al. obtained a recurrence relation representing the sequence A054548(OEIS) of labeled graphs on n ≥ 0 unisolated vertices containing q ≥ (n)/(2) edges. In 2020, Bhavale and Waphare obtained a recurrence relation representing the sequence of fundamental basic blocks on n ≥ 0 comparable reducible elements, having nullity l ≥ \lfloor (n+1)/(2) \rfloor. In this paper, we prove the equivalence of these two sequences. We also provide an edge labeling for a given vertex labeled finite simple graph.