On the characteristic polynomials of Dutch windmill graphs and their applications

Wenjing Li; Yiwei Zhang; Ying Wang; Wenjing Li; Yiwei Zhang; Ying Wang

doi:10.3934/math.2026685

AIMS Mathematics

2026, Volume 11, Issue 6: 16697-16711. doi: 10.3934/math.2026685

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On the characteristic polynomials of Dutch windmill graphs and their applications

School of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou 450046, China

Received: 27 December 2025 Revised: 25 May 2026 Accepted: 03 June 2026 Published: 11 June 2026
MSC : 05C50

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 $ \Phi_A(D_p^q, \lambda) $, the Laplacian polynomial $ \Phi_L(D_p^q, \mu) $, and the signless Laplacian polynomial $ \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.
- adjacency spectrum,
- Laplacian spectrum,
- signless Laplacian spectrum,
- graph energy,
- Laplacian energy,
- signless Laplacian energy,
- Dutch windmill graph
Citation: Wenjing Li, Yiwei Zhang, Ying Wang. On the characteristic polynomials of Dutch windmill graphs and their applications[J]. AIMS Mathematics, 2026, 11(6): 16697-16711. doi: 10.3934/math.2026685

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Abstract

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 $ \Phi_A(D_p^q, \lambda) $, the Laplacian polynomial $ \Phi_L(D_p^q, \mu) $, and the signless Laplacian polynomial $ \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.

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