2025/07/15 by Zhou, Sizhong · 2 citations
#05C05 #05C50 #05C70 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2507.11078
A spanning tree T of a connected graph G is a subgraph of G that is a tree covers all vertices of G. The leaf distance of T is defined as the minimum of distances between any two leaves of T. A fractional matching of a graph G is a function h assigning every edge a real number in [0,1] so that ∑e∈ EG(v)h(e)≤1 for any v∈ V(G), where EG(v) denotes the set of edges incident with v in G. A fractional matching of G is called a fractional perfect matching if ∑e∈ EG(v)h(e)=1 for any v∈ V(G). A graph G with at least 2k+2 vertices is said to be fractional k-extendable if every k-matching M in G is included in a fractional perfect matching h of G such that h(e)=1 for any e∈ M. This paper considers a lower bound on the spectral radius of G to guarantee that G has a spanning tree with leaf distance at least d. At the same time, we obtain a lower bound on the spectral radius of G to ensure that G is fractional k-extendable.