On distance Laplacian spectral radius and chromatic number of graphs

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Let G be a connected simple graph with n vertices having chromatic number χ. The distance Laplacian matrix DL(G) is defined as DL(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 DL(G) are the distance Laplacian eigenvalues of G and are denoted by ∂1L(G),∂2L(G),…,∂nL(G). The largest eigenvalue ∂1L(G) is called the distance Laplacian spectral radius. For a non-complete graph G with n vertices and chromatic number χ, Aouchiche and Hansen (2017) proved that ∂1L(G)≥n+⌈nχ⌉. If G is a connected graph with n≥4 vertices and chromatic number χ≤n−2, we prove that ∂2L(G)≥n+⌈nχ⌉ and we show the existence of graphs for which the equality holds. Among all graphs with chromatic number χ satisfying n2≤χ≤n−1, we show that the graph K2,2,…,2︸n−χ,1,1,…,1︸2χ−n has the minimum distance Laplacian spectral radius. Also, we give the distribution of the distance Laplacian eigenvalues in relation to the chromatic number χ and other graph invariants. We characterize the extremal graphs for some of these results and for others, we illustrate by examples.