Characterizing an odd [1, b]-factor on the distance signless Laplacian spectral radius

Let G be a connected graph of even order n. An odd [1,b]-factor of G is a spanning subgraph F of G such that dF(v) ∈ {1,3,5,··· ,b} for any v ∈ V (G), where b is positive odd integer. The distance matrix D(G) of G is a symmetric real matrix with (i,j)-entry being the distance between the vertices vi and vj. The distance signless Laplacian matrix Q(G) of G is defined by Q(G) = Tr(G) + D(G), where Tr(G) is the diagonal matrix of the vertex transmissions in G. The largest eigenvalue η1(G) of Q(G) is called the distance signless Laplacian spectral radius of G. In this paper, we verify sharp upper bounds on the distance signless Laplacian spectral radius to guarantee the existence of an odd [1,b]-factor in a graph; we provide some graphs to show that the bounds are optimal.