On Distribution of Laplacian Eigenvalues of Graphs

Abstract

The work in this thesis concerns the investigation of eigenvalues of the Laplacian matrix, normalized Laplacian matrix, signless Laplacian matrix and distance signless Laplacian matrix of graphs. In Chapter 1, we present a brief introduction of spectral graph theory with some definitions. Chapter $2$ deals with the sum of $ k $ largest Laplacian eigenvalues $ S_{k}(G) $ of graph $ G $ and Brouwer's conjecture. We obtain the upper bounds for $ S_{k}(G) $ for some classes of graphs and use them to verify Brouwer's conjecture for these classes of graphs. Also, we prove Brouwer's conjecture for more general classes of graphs. In Chapter $3$, we investigate the Laplacian eigenvalues of graphs and the Laplacian energy conjecture for trees. We prove the Laplacian energy conjecture completely for trees of diameter $ 4 $. Further, we prove this conjecture for all trees having at most $ \frac{9n}{25}-2 $ non-pendent vertices. Also, we obtain the sufficient conditions for the truth of conjecture for trees of order $ n $. In Chapter $4$, we determine the normalized Laplacian spectrum of the joined union of regular graphs and obtain the spectrum of some well known graphs. As consequences of joined union, we obtain the normalized Laplacian spectrum of power graphs associated to finite cyclic groups. In Chapter $5$, we find the distance signless Laplacian spectrum of regular graphs and zero-divisor graphs associated to finite commutative ring. Also, we find the bounds for spectral radius of generalized distance matrix. Further, we obtain the generalized distance energy for bipartite graphs and trees. We prove that the complete bipartite graph has minimum generalized distance energy among all connected bipartite graphs. Besides, for $ \alpha\in \big(0, \frac{2n}{3n-2}\big) $, we show that the star graph has minimum generalized distance energy among all trees.