On distance Laplacian spectral radius and chromatic number of graphs

Let G be a connected simple graph with n vertices having chromatic number chi. The distance Laplacian matrix D-L(G) is defined as D-L(G) = Diag(Tr) - D, where Diag(Tr) is the diagonal matrix of vertex transmissions and D is the distance matrix of G. The eigenvalues of D-L(G) are the distance Laplacian eigenvalues of G and are denoted by partial derivative(L)(1)(G), partial derivative(L)(2)(G), ..., partial derivative(L)(n)(G). The largest eigenvalue partial derivative(L)(1)(G) is called the distance Laplacian spectral radius. For a non-complete graph G with n vertices and chromatic number chi, Aouchiche and Hansen (2017) proved that partial derivative(L)(1)(G) >= n + inverted right perpendicular n/chi inverted left perpendicular. If G is a connected graph with n >= 4 vertices and chromatic number chi <= n - 2, we prove that partial derivative(L)(2)(G) >= n + inverted right perpendicular n/chi inverted left perpendicular and we show the existence of graphs for which the equality holds. Among all graphs with chromatic number chi satisfying n/2 <= chi <= n - 1, we show that the graph K2, 2, ..., 2}n-chi, 1, 1, ..., 1}2 chi-n has the minimum distance Laplacian spectral radius. Also, we give the distribution of the distance Laplacian eigenvalues in relation to the chromatic number chi and other graph invariants. We characterize the extremal graphs for some of these results and for others, we illustrate by examples. (C) 2021 Elsevier Inc. All rights reserved.