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The HyperZagreb Index and Some Properties of Graphs

Author(s): Rao Li (University of South Carolina at Aiken, USA)
Copyright: 2020
Pages: 15
Source title:
Handbook of Research on Advanced Applications of Graph Theory in Modern Society
Source Author(s)/Editor(s): Madhumangal Pal (Vidyasagar University, India), Sovan Samanta (Tamralipta Mahavidyalaya, India) and Anita Pal (National Institute of Technology Durgapur, India)
DOI: 10.4018/9781522593805.ch006
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Abstract
Let G = (V(G), E(G)) be a graph. The complement of G is denoted by Gc. The forgotten topological index of G, denoted F(G), is defined as the sum of the cubes of the degrees of all the vertices in G. The second Zagreb index of G, denoted M2(G), is defined as the sum of the products of the degrees of pairs of adjacent vertices in G. A graph GiskHamiltonian if for all X ⊂V(G) withX ≤ k, the subgraph induced byV(G)  Xis Hamiltonian. Clearly, G is 0Hamiltonian if and only if G is Hamiltonian. A graph GiskpathcoverableifV(G) can be covered bykor fewer vertexdisjoint paths. Using F(Gc) and M2(Gc), Li obtained several sufficient conditions for Hamiltonian and traceable graphs (Rao Li, Topological Indexes and Some Hamiltonian Properties of Graphs). In this chapter, the author presents sufficient conditions based upon F(Gc) and M2(Gc)for kHamiltonian, kedgeHamiltonian, kpathcoverable, kconnected, and kedgeconnected graphs.
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