Equivalence of labeled graphs and lattices

TL;DR

This paper proves the equivalence between two sequences related to labeled graphs and fundamental blocks, and provides a method for edge labeling in labeled finite simple graphs.

Contribution

It establishes the equivalence of two previously studied graph sequences and introduces an edge labeling technique for labeled finite simple graphs.

Findings

Proved the equivalence of two graph sequences.

Provided an edge labeling method for labeled graphs.

Connected graph enumeration problems from different perspectives.

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.