Atom bond connectivity index for graph with self-loops and its application to structure property relationships in anticancer drugs
B. Sharath, H. J. Gowtham
Abstract
Let
$$G_S$$
be a graph derived from a simple graph G by adding a self-loop to each vertex in a subset
$$S\subseteq V(G)$$
. In this paper, we define the atom bond connectivity index of the graph
$$G_S$$
as
$$ABC(G_S)$$
and the atom bond connectivity energy of
$$G_S$$
as
$$E_{ABC}(G_S)$$
. We obtained upper bounds for the ABC spectral radius of the graph
$$G_S$$
as well as bounds for
$$E_{ABC}(G_S)$$
and
$$ABC(G_S)$$
in terms of m, n,
$$\Delta$$
and
$$\delta$$
. Additionally, we computed the ABC energy for complete graph, cocktail party graph and crown graph with self-loops. We also derived the characteristic polynomial of double star graph with self-loops. Furthermore, we explored the correlation between
$$ABC(G_S)$$
and various physico-chemical properties, such as boiling point (BP) and molar refraction (MR). Furthermore, we established correlations between
$$ABC(G_S)$$
and specific indices, specifically the Sombor index of a graph
$$G_S$$
$$(SO(G_S))$$
, first Zagreb index of a graph
$$G_S$$
$$(M_1(G_S))$$
, and Randic index of a graph
$$G_S$$
$$(R(G_S))$$
.