Abstract
For a graph \( G \) with vertex set \( V(G) \) and edge set \( E(G) \), the Lanzhou index of \( G \) is defined as \[ Lz(G)=\sum\limits_{\upsilon{\in}V(G)}d_{G}(\upsilon)^2{d_{\overline{G}}(\upsilon)}, \] where \( d_{G}(\upsilon) \) is the degree of vertex
\( \upsilon \) in \( G \), \( {\overline{G}} \) is the complement of \( G \). Vukičević, Li, Sedlar and Došlić [MATCH Commun. Math. Comput. Chem. 86 (2021) 3-10] proved that for any tree \( T \) of order \( n \geq 11 \) with maximum degree
\( \Delta \), \( Lz(T) \geq (n-\Delta-1)(4n+{\Delta}^2-12)+\Delta(n-2) \). In this paper, we generalize the foregoing bound and we show that for any unicyclic graph \( U \) of order \( n\geq 11 \) with maximum degree \( \Delta \), \( Lz(U)\geq4(n-3)(n-\Delta
+1)+\Delta^2(n-1-\Delta)+(n-2)(\Delta-2) \), and we also characterize the corresponding extremal unicyclic graphs.
