The Lanzhou Index of Unicyclic Graphs with Fixed Maximum Degree

For a graphGwith vertex setV(G) and edge setE(G), theLanzhou index ofGis defined asLz(G) =X upsilon is an element of V(G)dG(upsilon)2dG(upsilon),wheredG(upsilon) is the degree of vertex upsilon inG,Gis the complementofG. Vukicevi c, Li, Sedlar and Dosli c [MATCH Commun. Math.Comput. Chem. 86 (2021) 3-10] proved that for any treeTof ordern >= 11 with maximum degree triangle,Lz(T)>=(n-triangle-1)(4n+ triangle 2-12) + triangle(n-2). In this paper, we generalize the foregoing boundand we show that for any unicyclic graphUof ordern >= 11 withmaximum degree triangle,Lz(U)>= 4(n-3)(n-triangle + 1) + triangle 2(n-1-triangle) +(n-2)(triangle-2), and we also characterize the corresponding extremalunicyclic graphs.