Equivalence of labeled graphs and lattices
Ashok Nivrutti Bhavale
TL;DR
The paper addresses the equivalence between two enumeration frameworks for labeled graphs and dismantlable lattices. It develops an explicit edge-labeling scheme for vertex-labeled graphs and constructs complete fundamental blocks CF$(n)$ with adjunct representations, enabling a bijection between fundamental blocks on $n$ reducible elements with nullity $l$ and directed graphs on $n$ vertices with $l$ edges. This yields a direct proof that the sequences $f(n,l)$ and $d(n,l)$ are identical for $\lfloor (n+1)/2\rfloor \le l \le \binom{n}{2}$, and hence aligns with the OEIS sequence $A054548$. The results provide a unified combinatorial framework linking labeled graph enumeration to lattice-theoretic constructions via adjunction, with a concrete edge-labeling method that uniquely determines edges from vertex labels. This equivalence has implications for understanding and computing graph counts across related combinatorial structures.
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.