The maximum PI index of bicyclic graphs with even number of edges

doi:10.1016/j.ipl.2019.02.001

Information Processing Letters

Volume 146, June 2019, Pages 13-16

https://doi.org/10.1016/j.ipl.2019.02.001 Get rights and content

Abstract

The PI index of a graph G is defined by $P I (G) = \sum_{e = (u, v) \in E} [m_{u} (e | G) + m_{v} (e | G)]$ where $m_{u} (e | G)$ be the number of edges in G lying closer to the vertex u than to the vertex v. In this paper, we give the upper bound on the PI index of connected bicyclic graphs with even number of edges and characterize the extremal graphs with maximum PI index.

Section snippets

Introduction and background

Let $G = (V, E)$ be a simple connected graph with $n = | V |$ vertices and $m = | E |$ edges. For more notations and terminologies that will be used, see [6].

A topological index is a real number related to a graph. It must be a structural invariant, i.e., it preserves by every graph automorphism. Several topological indices have been defined and many of them have found applications as means to model chemical, pharmaceutical and other properties of molecules.

For each edge $e = (u, v) \in E$ , let $m_{u} (e | G)$ be the number of

The proof of the main results

In this section, we give the proof of Theorem 1. Let G be a connected bicyclic graph with m edges where m is even throughout this section. We just need to prove that $S^{⁎} - m \geq 5$ . We consider the following two cases.

References (28)

G. Ma et al.
Correction of the paper “Bicyclic graphs with extremal values of PI index”
Discrete Appl. Math.
(2016)
G. Ma et al.
Bounds on the PI index of unicyclic and bicyclic graphs with given girth
Discrete Appl. Math.
(2017)
Ž.K. Vukićević et al.
Bicyclic graphs with extremal values of PI index
Discrete Appl. Math.
(2013)
H. Yousefi-Azari et al.
The PI index of product graphs
Appl. Math. Lett.
(2008)
A.R. Ashrafi et al.
Computing PI and Omega polynomials of an infinite family of fullerenes
MATCH Commun. Math. Comput. Chem.
(2008)
A.R. Ashrafi et al.
PI index of armchair polyhex nanotubes
Ars Comb.
(2006)
A.R. Ashrafi et al.
PI index of zig–zag polyhex nanotubes
MATCH Commun. Math. Comput. Chem.
(2006)
A.R. Ashrafi et al.
On the PI polynomial of a graph
Util. Math.
(2006)
A.R. Ashrafi et al.
PI index of polyhex nanotori
MATCH Commun. Math. Comput. Chem.
(2007)
J.A. Bondy et al.
Graph Theory
(2008)

H. Deng

The PI index of $T U V C_{6} [2 p, q]$

MATCH Commun. Math. Comput. Chem.

(2006)

H. Deng

On the PI index of a graph

MATCH Commun. Math. Comput. Chem.

(2008)

H. Deng

Extremal catacondensed hexagonal systems with respect to the PI index

MATCH Commun. Math. Comput. Chem.

(2006)

H. Deng et al.

The PI index of phenylenes

J. Math. Chem.

(2007)

Cited by (2)

Algorithms for enumerating multiple leaf-distance granular regular α-subtree of unicyclic and edge-disjoint bicyclic graphs
2024, Applied Mathematics and Computation
A multiple leaf-distance granular regular α-tree (abbreviated as LDR α-tree for short) is a tree (with at least $α + 1$ vertices) where any two leaves are at some distance divisible by α. A connected graph's subtree which is additionally an LDR α-tree is known as an LDR α-subtree. Obviously, $α = 1$ and 2, correspond to the general subtrees (excluding the single vertex subtrees) and the BC-subtrees (the distance between any two leaves of the subtree is even), respectively. With generating functions and structure decomposition, in this paper, we propose algorithms for enumerating an auxiliary subtree $α_{τ} (v)$ -subtree $(τ = 0, 1, \dots, α - 1)$ containing a fixed vertex, and various LDR α-subtrees of unicyclic graphs, respectively. Basing on these algorithms, we further present algorithms for enumerating various LDR α-subtrees of edge-disjoint bicyclic graphs.
Bounds on the weighted vertex pi index of cacti graphs
2019, Filomat

^☆: This work is supported by a research grant NSFC (11461054) of China.

View full text

The maximum PI index of bicyclic graphs with even number of edges☆

Abstract

Section snippets

Introduction and background

The proof of the main results

Discrete Appl. Math.

Discrete Appl. Math.

Discrete Appl. Math.

Appl. Math. Lett.

Computing PI and Omega polynomials of an infinite family of fullerenes

MATCH Commun. Math. Comput. Chem.

PI index of armchair polyhex nanotubes

Ars Comb.

PI index of zig–zag polyhex nanotubes

MATCH Commun. Math. Comput. Chem.

On the PI polynomial of a graph

Util. Math.

PI index of polyhex nanotori

MATCH Commun. Math. Comput. Chem.

Graph Theory

The PI index of TUVC6[2p,q]

MATCH Commun. Math. Comput. Chem.

On the PI index of a graph

MATCH Commun. Math. Comput. Chem.

Extremal catacondensed hexagonal systems with respect to the PI index

MATCH Commun. Math. Comput. Chem.

The PI index of phenylenes

J. Math. Chem.

The PI index of $T U V C_{6} [2 p, q]$