45 research outputs found
Cacti with Extremal PI Index
The vertex PI index is a
distance-based molecular structure descriptor, where denotes the
number of vertices which are closer to the vertex than to the vertex
and which has been the considerable research in computational chemistry dating
back to Harold Wiener in 1947. A connected graph is a cactus if any two of its
cycles have at most one common vertex. In this paper, we completely determine
the extremal graphs with the largest and smallest vertex PI indices among all
the cacti. As a consequence, we obtain the sharp bounds with corresponding
extremal cacti and extend a known result.Comment: Accepted by Transactions on Combinatorics, 201
On Topological Indices And Domination Numbers Of Graphs
Topological indices and dominating problems are popular topics in Graph Theory. There are various topological indices such as degree-based topological indices, distance-based topological indices and counting related topological indices et al. These topological indices correlate certain physicochemical properties such as boiling point, stability of chemical compounds. The concepts of domination number and independent domination number, introduced from the mid-1860s, are very fundamental in Graph Theory. In this dissertation, we provide new theoretical results on these two topics. We study k-trees and cactus graphs with the sharp upper and lower bounds of the degree-based topological indices(Multiplicative Zagreb indices). The extremal cacti with a distance-based topological index (PI index) are explored. Furthermore, we provide the extremal graphs with these corresponding topological indices. We establish and verify a proposed conjecture for the relationship between the domination number and independent domination number. The corresponding counterexamples and the graphs achieving the extremal bounds are given as well
Cacti with minimum, second-minimum, and third-minimum Kirchhoff indices
Resistance distance was introduced by Klein and
RandiÄ. The Kirchhoff index Kf(G) of a graph G is the
sum of resistance distances between all pairs of vertices. A graph
G is called a cactus if each block of G is either an edge or a
cycle. Denote by Cat(n; t) the set of connected cacti possessing
n vertices and t cycles. In this paper, we give the first three
smallest Kirchhoff indices among graphs in Cat(n; t), and
characterize the corresponding extremal graphs as well
Graphical Indices and their Applications
The biochemical community has been using graphical (topological, chemical) indices in the study of Quantitative Structure-Activity Relationships (QSAR) and Quantitative Structure-Property Relationships (QSPR), as they have been shown to have strong correlations with the chemical properties of certain chemical compounds (i.e. boiling point, surface area, etc.). We examine some of these chemical indices and closely related pure graph theoretical indices: the RandiÄ index, the Wiener index, the degree distance, and the number of subtrees. We find which structure will maximize the RandiÄ index of a class of graphs known as cacti, and we find a functional relationship between the Wiener index and the degree distance for several types of graphs. We also develop an algorithm to find the structure that maximizes the number of subtrees of trees, a characterization of the second maximal tree may also follow as an immediate result of this algorithm
Degree Distance and Gutman Index of Two Graph Products
The degree distance was introduced by Dobrynin, Kochetova and Gutman as a weighted version of the Wiener index. In this paper, we investigate the degree distance and Gutman index of complete, and strong product graphs by using the adjacency and distance matrices of a graph
On the Gutman index of Thorn graphs
In this paper, the relation between the Gutman index of a simple connected graph and its thorn graph is stablished and several special cases of the result are examined. Results are applied to compute the Gutman index of thorn paths, thorn rods, caterpillars, thorn rings, thorn stars, Kragujevac trees, and dendrimers