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

Xiangge Liu (), Yong Lu () and Qiannan Zhou ()
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Xiangge Liu: Jiangsu Normal University
Yong Lu: Jiangsu Normal University
Qiannan Zhou: Jiangsu Normal University

Journal of Combinatorial Optimization, 2025, vol. 49, issue 5, No 25, 20 pages

Abstract: Abstract 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.

Keywords: Hamiltonian; Lanzhou index; Ad-hoc Lanzhou index; Sufficient condition; 05C35; 05C50 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s10878-025-01328-w

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