Express the number of spanning trees in term of degrees

• Published in: Applied mathematics and computation Vol. 415; p. 126697
• Main Authors: Dong, Fengming, Ge, Jun, Ouyang, Zhangdong
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.02.2022
• Subjects: Degree; Graph polynomial; Spanning tree; Graph polynomial; Degree; Spanning tree; 05C30; 05C05
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2021.126697

•Page 2, the first line below Theorem 1: The set NSTu(T) is revised to the set of non-spanning trees T such that u∈V(T) and G−V(T) has no isolated vertices.•Add two lemmas in the beginning of Section 3. The purpose is to apply them to simplify the proof of Theorem 3.•The proof of Theorem 3 has been revised according to the suggestion from one of the reviewers to consider the set of spanning digraphs D which have the property that idD(vn)=0 and idD(vi)=1 for each i∈[n−1].•In page 8, we add a paragraph for an upper bound of τ(G) and a corollary. It is well-known that the number of spanning trees, denoted by τ(G), in a connected multi-graph G can be calculated by the Matrix-Tree Theorem and Tutte’s deletion-contraction formula. In this short note, we find an alternate method to compute τ(G) by degrees of vertices.