Abstract
Let \(G\) be a finite and simple graph with vertex set \(V(G)\). The Lanzhou index \(G\) is defined as \[ L_z(G) =\sum_{u\in V(G)}d_{\overline {G}}(u)d_G(u)^2; \] where \(d_G(u)\) denotes the degree of vertex \(u\) in \(G\). Dehgardi and Liu [MATCH Commun.
Math. Comput. Chem. 86 (2021) 3--10] proved that for any tree \(T\) of order \)n \geq 11\) with maximum degree \(\Delta\), \(L_z(T) \geq (n-\Delta-1)(4n+\Delta^2-12)+\Delta(n-2).\) In this paper, we generalize the foregoing bound and show
that for non-spider tree \(T\) of order \(n\geq 11\) \(L_z(T)\geq (n-1)(\Delta^2+\Delta'^2)-(\Delta^3+\Delta'^3)-(3n-10)(\Delta+\Delta')+(4n^2-14n+4),\) where \(\Delta\) and \(\Delta'\) represents the maximum and second maximum degree
of \(T\). This result is an improvement of existing lower bounds. We also characterize the corresponding extremal trees.