Vertex PI v Topological Index of Titania Carbon Nanotubes TiO 2 (m,n)

The Padmakar – Ivan (PI) index of polyominoes is examined.Efficient calculations of formulas for PI index for the polyominoes are put forward. In chemical graph theory, the PI index is a topological indexof a graph G is defined as , where for edge e = xy, n1 (e) is the number of edges of G lying closer to x than y, n2 (e) is the number of edges of G lying closer to y than x and summation goes over all edges of G. The edges equidistant from x and y are not considered for the calculation of PI index. In this paper, we calculated the PI index of polyominoes like square Polyomino, L-Polyomino,T-Polyomino, Straight-Polyomino and Skew-Polyomino.

The Padmakar-Ivan (PI) index of a graph G is defined as PI(G) = ∑(n eu (e|G)+ n ev (e|G)), where n eu (e|G) is the number of edges of G lying closer to u than to v, n ev (e|G) is the number of edges of G lying closer to v than to u and summation goes over all edges of G. In this paper, we first compute the PI index of a class of pericondensed benzenoid graphs consisting of n rows, n ≤ 3, of hexagons of various lengths. Finally, we prove that for any connected graph G with exactly m edges, PI(G) ≤ m(m-1) with equality if and only if G is an acyclic graph or a cycle of odd length. 1 . Here, we consider a new topological index, named the Padmakar-Ivan index, which is abbreviated as the PI index 2-17 . This newly proposed topological index, differ from the Wiener index 18 , the oldest topological index for acyclic (tree) molecules. We now describe some notations which will be adhered to throughout. Benzenoid systems (graph representations of benzenoid hydrocarbons) are defined as finite connected plane graphs with no cut-vertices, in which all interior regions are mutually congruent regular hexagons. More details on this important class of molecular graphs can be found in the book of Gutman and Cyvin 19 and in the references cited therein. Let G be a simple molecular graph without directed or multiple edges and without loops, the vertex and edge-shapes of which are represented by V(G) and E(G), respectively. The graph G is said to be connected if for every pair of vertices x and y in V(G) there exists a path between x and y. In this paper we only consider connected graphs. If e is an edge of G, connecting the vertices u and v then we write e=uv. The number of vertices of G is denoted by n. The distance between a pair of vertices u and w of G is denoted by d(u,w). We now define the PI index of a graph G. To do this, suppose that e = uv and introduce the quantities neu(e|G) and nev(e|G). neu(e|G) is the number of edges lying closer to vertex u than to vertex v, and nev(e|G) is the number of edges lying closer to vertex v than to vertex u. Then PI(G) = ∑(neu(e|G) + nev(e|G)), where the summation goes over all edges of G. Edges equidistant from both ends of the edge e = uv are not counted and the number of such edges is denoted by N(e). To clarify this, for every vertex u and any edge f = zw of graph G, we define d(f,u) = Min{d(u,w),d(u,z)}. Then f is equidistant from both ends of the edge e = uv if d(f,u) = d(f,v). In a series of papers, Khadikar and coauthors 2-17 defined and then computed the PI index of some chemical graphs. The present author 20 computed the PI index of a zig-zag polyhex nanotube. In this paper we continue this study to prove an important result concerning the PI index and find an exact expression for the PI index of some other chemical graphs. Our notation is standard and mainly taken from the literature. 21,22

The Padmakar-Ivan (PI) index of a graph G is given by PI ( G ) = ∑ e ∈ E ( G ) ( | V ( G ) | − N G ( e ) ) , where N G ( e ) is the number of equidistant vertices for the edge e. This paper presents a triangle cover for a graph, along with a novel method for finding the PI index using this cover. The technique is used to examine some chemical networks, including octahedral and oxide networks, leading to the determination of the exact formula for their PI indices. Additionally, the approach is used to study perfect graphs such as prismatic and chordal graphs. Finally, the PI index of chordal graphs with a diameter of two is investigated, focusing on their induced subgraphs.

In this survey article a brief account on the development of Padmakar-Ivan (PI) index in that applications of Padmakar-Ivan (PI) index in the fascinating field of nano-technology are discussed.

The Padmakar-Ivan (PI) index of a graph G is defined as PI(G) = ∑[neu(e|G) + nev(e|G)], where neu(e|G) is the number of edges of G lying closer to u than to v, nev(e|G) is the number of edges of G lying closer to v than to u and summation goes over all edges of G. The PI Index is a Szeged-like topological index developed very recently. In this paper we report on new results about computing PI index of nanotubes.

QSAR Study on solubility of alkanes in water and Their partition coefficients in different solvent systems using PI index

The Padmakar-Ivan (PI) index is a Wiener-Szeged-like topological index which reflects certain structural features of organic molecules. The PI index of a graph G is the sum of all edges uv of G of the number of edges which are not equidistant from the vertices u and v. In this paper we obtain the second and third extremals of catacondensed hexagonal systems with respect to the PI index.

QSAR study on bioconcentration factor (BCF) of polyhalogented biphenyls using the PI index

The Padmakar–Ivan (PI) index of a graph G is defined as PI $$(G) = \Sigma [n_{\rm eu}(e\vert G)+n_{\rm ev}(e\vert G)]$$ , where for edge e=(u,v) are $$n_{\rm eu} (e\vert G)$$ the number of edges of G lying closer to u than v, and $$n_{\rm ev} (e\vert G)$$ is the number of edges of G lying closer to v than u and summation goes over all edges of G. The PI index is a Wiener–Szeged-like topological index developed very recently. In this paper, we describe a method of computing PI index of benzenoid hydrocarbons (H) using orthogonal cuts. The method requires the finding of number of edges in the orthogonal cuts in a benzenoid system (H) and the edge number of H – a task significantly simpler than the calculation of PI index directly from its definition.

On the extremal graphs with respect to the vertex PI index

QSAR study on toxicity to aqueous organisms using the PI index

<p>Department of Mathematics, Statistics and Computer Science, Faculty of Science, University of Kashan.</p>\n\n<p>Kashan 87317-51167, I. R. Iran</p>\n\n<p><em>Manuscript received 5 October 2010, accepted 6 October 2010</em></p>\n\n<p>The Padmakar-lvan (PI) index of a graph <em>G</em> is defined as PI(<em>G</em>) = ∑<sub> e=uv</sub>[<em>m<sub>u</sub>(e) + m<sub>v</sub>(e</em>)], where <em>m<sub>u</sub> (e)</em> is the number of edges of G lying closer to <em>u</em> than to <em>v, m<sub>v</sub>(e</em>) is defined analogously and summation goes over all edges of G. In this paper, the PI index of tetrathiafulvalene (TTF) dendrimer Is computed, whereby TTF units were employed as branching centers.</p>

Vertex and edge PI indices of Cartesian product graphs

Graph theory, introduced by the Swiss mathematician Leonhard Euler in 1736, has played a pivotal role in solving real-world problems since its inception, notably exemplified by Euler's solution to the Konigsberg Bridge problem. Its applications extend to various domains, including scheduling, shortest path routing, and chemical structure representation. In chemistry, graphs are extensively used to depict molecular structures and chemical compounds, aiding in visualizing atomic connections and overall compound configurations. Topology indices, such as the Padmakar-Ivan (PI) and Randic indices, provide numerical values capturing chemical bonding relationships. Beyond chemical structures, these indices find applications in abstract algebraic graph representations. Recent research, exemplified by Husni et al.'s work on the harmonic and Gutman indices, explores these indices in coprime graphs of integer groups modulo prime power orders. Additionally, studies on non-coprime graphs of integer groups modulo reveal unique characteristics and invariants, shedding light on their structure. The non-coprime graph is a graph with two vertices said to be adjacent if the greatest common divisor (GCD) of their orders is not equal to one. This paper aims to investigate the topological indices, specifically the Padmakar-Ivan and Randic indices, in non-coprime graphs of integer groups modulo, adding depth to our understanding of their applicability and significance in abstract algebraic representations.
 Keywords: Graph theory, padmakar-ivan index, randic index, non-coprime graphsMSC2020: 05C09

Correlations between the benzene character of acenes or helicenes and simple molecular descriptors