Eigenvalues and spanning tree in regular graphs

Let k ≥ 2 be an integer and let T be a spanning tree of a connected graph. The total k -excess t e ( T , k ) of T is the summation of the k -excesses of all vertices in T , namely, t e ( T , k ) = ∑ v ∈ V ( T ) max { 0 , d T ( v ) − k } . One can see that T is a spanning k -tree if and only if t e ( T , k ) = 0 . Recently, O [J. Graph Theory 100 (2022) 458-469] proposed a meaningful and intriguing problem: What is a best possible upper bound on a certain eigenvalue which can guarantee a d -regular graph to contain a connected [ a , b ] -factor, where a and b are two positive integers with a ≤ b ? Note that a spanning tree with t e ( T , k ) ≤ t is a connected [ 1 , k + t ] -factor, where t ≥ 0 is an integer. Let λ i ( G ) be the i -th largest eigenvalue of a graph G . Focusing on the problem due to O and adopting interlacing theorems, we in this paper prove an upper bound on λ t + 4 ( G ) to guarantee a connected d -regular graph to contain a spanning tree with t e ( T , k ) ≤ t . As a corollary, we derive an upper bound on λ 4 ( G ) for the existence of a spanning k -tree in a connected d -regular graph G , which improves the result of Wong. Moreover, we also prove an upper bound on λ k + t + 1 ( G ) to guarantee a connected d -regular graph G to contain a spanning tree with t e ( T , k ) ≤ t .