Equivalence of labeled graphs and lattices

Abstract

In $1973$, Harary and Palmer posed the problem of enumeration of labeled graphs on $n \geq 1$ unisolated vertices and $l \geq 0$ edges. In $1997$, Bender et al.\ obtained a recurrence relation representing the sequence $A054548$(OEIS) of labeled graphs on $n \geq 0$ unisolated vertices containing $q \geq \frac{n}{2}$ edges. In $2020$, Bhavale and Waphare obtained a recurrence relation representing the sequence of fundamental basic blocks on $n \geq 0$ comparable reducible elements, having nullity $l \geq \lfloor \frac{n+1}{2} \rfloor$. 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.