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.

Let \(G\) be a graph with vertex set \(V\) and edge set \(E\) such that every edge \(e\in E\) belongs to at least one copy of a given subgraph \(H\) of \(G\). A bijection \(f:V\cup E\to \{1,2,\dots,|V|+|E|\}\) is called an \(H\)-supermagic labeling if the sum of labels of all vertices and edges of every copy of \(H\) is equal to the same number \(\mu\) and the vertices are labeled with the first \(|V|\) integers. A \(p\)-calendula graph \(Cal_{m,p[n]}\) consists of a cycle \(C_m\) with \(p\) copies of \(C_n\) amalgamated to each edge of \(C_m\). We generalize a previous result by Pradipta and Salman on 1-calendula graphs by providing \(C_n\)-supermagic labelings of \(Cal_{m,p[n]}\) for all \(m,n\geq3, p\geq 1\), and \(m\neq n\). The case of \(m=n, \ p>1\) remains open.

In this note, we present sufficient conditions involving the independence number, the minimum degree, and the maximum degree for some Hamiltonian properties of a graph.