Alpha labelings of disjoint union of hairy cycles

In this paper, we prove the following results: (1) the disjoint union of $n\geq 2$ isomorphic copies of a graph obtained by adding a pendant edge to each vertex of a cycle of order $4$ admits an $\alpha$-valuation; (2) the disjoint union of two isomorphic copies of a graph obtained by adding $n\geq 1$ pendant edges to each vertex of a cycle of order $4$ admits an $\alpha$-valuation; (3) the disjoint union of two isomorphic copies of a graph obtained by adding a pendant edge to each vertex of a cycle of order $4m$ admits an $\alpha$-valuation; (4) the disjoint union of two nonisomorphic copies of a graph obtained by adding a pendant edge to each vertex of cycles of order $4m$ and $4m-2$ admits an $\alpha$-valuation; (5) the disjoint union of two isomorphic copies of a graph obtained by adding a pendant edge to each vertex of a cycle of order $4m-1$ $(4m+2)$ admits a graceful valuation (an $\alpha$-valuation), respectively.