Equivalence of labeled graphs and lattices

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