Sufficient conditions for some graphical properties in terms of the Lanzhou index and the ad-hoc Lanzhou index

During the past decade, several research groups have published sufficient conditions for Hamiltonicity of graphs in terms of the first Zagreb index, the second Zagreb index and the forgotten topological index. The forgotten topological index (F-index) is defined as $$F(G)=\sum \limits _{uv\in E(G)}(d^{2}(u)+d^{2}(v))=\sum \limits _{v\in V(G)}d^{3}(v)$$ F ( G ) = ∑ u v ∈ E ( G ) ( d 2 ( u ) + d 2 ( v ) ) = ∑ v ∈ V ( G ) d 3 ( v ) . The forgotten topological coindex (F-coindex) is defined as $${\overline{F}}(G)=\sum \limits _{uv\notin E(G)}(d^{2}(u)+d^{2}(v))=\sum \limits _{v\in V(G)}d^{2}(v)(n-d(v)-1)$$ F ¯ ( G ) = ∑ u v ∉ E ( G ) ( d 2 ( u ) + d 2 ( v ) ) = ∑ v ∈ V ( G ) d 2 ( v ) ( n - d ( v ) - 1 ) and it can be also called the Lanzhou index Lz(G). The Lanzhou index of the complement of G is the ad-hoc Lanzhou index and defined as $$\widetilde{Lz}(G)=\sum \limits _{v\in V(G)}d(v)(n-d(v)-1)^{2}$$ Lz ~ ( G ) = ∑ v ∈ V ( G ) d ( v ) ( n - d ( v ) - 1 ) 2 . This paper mainly focuses on sufficient conditions for graphs to be traceable, Hamiltonian, Hamilton-connected, k-path-coverable, k-Hamiltonian, k-edge-Hamiltonian and k-leaf-connected in terms of the Lanzhou index and the ad-hoc Lanzhou index.