Symmetry (Sep 2023)

On Laplacian Eigenvalues of Wheel Graphs

  • Manal Alotaibi,
  • Ahmad Alghamdi,
  • Hanan Alolaiyan

DOI
https://doi.org/10.3390/sym15091737
Journal volume & issue
Vol. 15, no. 9
p. 1737

Abstract

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Consider G to be a simple graph with n vertices and m edges, and L(G) to be a Laplacian matrix with Laplacian eigenvalues of μ1,μ2,…,μn=zero. Write Sk(G)=∑i=1kμi as the sum of the k-largest Laplacian eigenvalues of G, where k∈{1,2,…,n}. The motivation of this study is to solve a conjecture in algebraic graph theory for a special type of graph called a wheel graph. Brouwer’s conjecture states that Sk(G)≤m+k+12, where k=1,2,…,n. This paper proves Brouwer’s conjecture for wheel graphs. It also provides an upper bound for the sum of the largest Laplacian eigenvalues for the wheel graph Wn+1, which provides a better approximation for this upper bound using Brouwer’s conjecture and the Grone–Merris–Bai inequality. We study the symmetry of wheel graphs and recall an example of the symmetry group of Wn+1, n≥3. We obtain our results using majorization methods and illustrate our findings in tables, diagrams, and curves.

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