On adjacent eccentric distance sum index

Abstract

For a connected graph $H$,
the adjacent eccentric distance sum index (AEDSI) is defined as
$$\xi^{sv}(H)=\sum_{v_{x}\in V(H)}\frac{\varepsilon_{H}(v_{x})\cdot
D_{H}(v_{x})}{\text{deg}_{H}(v_{x})},$$
where $\varepsilon_{H}(v_{x})$ denotes the eccentricity of the vertex $v_{x}$,
$\text{deg}_{H}(v_{x})$ is the degree of $v_{x}$ and
$D_{H}(v_{x})=\Sigma_{v_{y}\in V(H)}d(v_{x},v_{y})$ is the
sum of all distances from $v_{x}$ in
$H$. AEDSI is proven to be very helpful on predicting anti-HIV
activity. In this paper, we give a best possible upper bound on the
AEDSI of $H$ with given radius that guarantees $H$ is $\omega$-connected,
$\beta$-deficient, $\omega$-Hamiltonian,
$\omega$-path-coverable and $\omega$-edge-Hamiltonian, respectively.
This supplies a continuation of the results presented by Feng et al. (2017).