Affiliation:
1. Department of Mthematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
2. Department of Mathematical Sciences, Faculty of Applied Sciences, Umm Al-Qura University, P.O. Box 14035, Makkah 21955, Saudi Arabia
Abstract
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.
Subject
Physics and Astronomy (miscellaneous),General Mathematics,Chemistry (miscellaneous),Computer Science (miscellaneous)
Reference35 articles.
1. Dharwadker, A., and Pirzada, S. (2011). Graph Theory, Institute of Mathematics. [3rd ed.].
2. On the sum of the Laplacian eigenvalues of a graph and Brouwer’s conjecture;Ganie;Linear Algebra Appl.,2016
3. The grone-merris conjecture;Bai;Trans. Am. Math. Soc.,2011
4. Brouwer, A., and Haemers, W. (2012). Spectra of Graph. Personal. Individ. Differ., xiv+250.
5. Spectral threshold dominance, Brouwer’s conjecture and maximality of Laplacian energy;Trevisan;Linear Algebra Appl.,2017