On Hamiltonian properties of bipartite graphs and several topological indices

Abstract

For a connected graph $H$, the first Zagreb index $M_{1}(H)$ is equal to the sum of squares of
the degrees of the vertices of $H$.
The reciprocal degree distance of $H$, denoted by $\text{RDD}(H)$, is defined as
$$\text{RDD}(H)=\sum_{x\neq y}\frac{\text{deg}_{H}(x)+ \text{deg}_{H}(y)}{\text{dist}_{H}(x,y)},$$
where
$\text{deg}_{H}(x)$ is the degree of the vertex $x$ in $H$ and
$\text{dist}_{H}(x,y)$ denotes the distance between two vertices $x$ and $y$
in $H$. The forgotten topological index $F(H)$ of $H$ is
the sum of cubes of all its vertex degrees. In this paper, we give
a best possible lower bound on $M_{1}(H)$, $\text{RDD}(H)$ or $F(H)$
to ensure that a bipartite graph $H$ is Hamiltonian.