On the characteristic polynomials of Dutch windmill graphs and their applications

The Dutch windmill graph $D_p^q$ is formed by $q$ cycles of length $p$ sharing a common vertex $v_0$. In this paper, we derive closed-form expressions for the characteristic polynomials—specifically, the adjacency polynomial ${\mathrm{\Phi }}_A(D_p^q,\lambda )$, the Laplacian polynomial ${\mathrm{\Phi }}_L(D_p^q,\mu )$, and the signless Laplacian polynomial ${\mathrm{\Phi }}_{L^{+}}(D_p^q,\nu )$ —of this family of graphs. As a direct consequence, we compute the exact values of the graph energy, Laplacian energy, and signless Laplacian energy of Dutch windmill graphs.