A fractional matching of a graph G is a function f:E(G) →[0, 1] such that for each vertex v, Σe∈ΓG(v) f (e) ≤ 1. The fractional matching number of G is the maximum value of Σe∈E(G) f (e) over all fractional matchings f. Tian et al. (Linear Algebra Appl 506:579-587, 2016) determined the extremal graphs with minimum distance Laplacian spectral radius among n-vertex graphs with given matching number. However, a natural problem is left open:among all n-vertex graphs with given fractional matching number, how about the lower bound of their distance Laplacian spectral radii and which graphs minimize the distance Laplacian spectral radii? In this paper, we solve these problems completely.
Jin Yan, Yan Liu, Xue-Li Su
. Lower Bounds of Distance Laplacian Spectral Radii of n-Vertex Graphs in Terms of Fractional Matching Number[J]. Journal of the Operations Research Society of China, 2023
, 11(1)
: 189
-196
.
DOI: 10.1007/s40305-021-00358-5
[1] Aouchiche, M., Hansen, P.:Two Laplacians for the distance matrix of a graph.Linear Algebra Appl.439, 21-33(2013)
[2] Lin, H.Y., Zhou, B.:On the distance Laplacian spectral radius of graphs.Linear Algebra Appl.475, 265-275(2015)
[3] Lin, H., Zhou, B.:The changes in distance Laplacian spectral radius of graphs resulting from graft transformations.Discrete Appl.Math.219, 147-157(2017)
[4] Niu, A., Fan, D., Wang, G.:On the distance Laplacian spectral of bipartite graphs.Discrete Appl.Math.186, 207-213(2015)
[5] Tian, F., Wong, D., Ma, X.B.:Lower bounds of distance Laplacian spectral radii of n-vertex graphs in terms of matching number.Linear Algebra Appl.506, 579-587(2016)
[6] Scheinerman, E.R., Ullman, D.H.:Fractional Graph Theory.Wiley, New York (1997)
