[PPT] - On some topological upper bounds of the apex trees Sarfraz Ahmad PowerPoint Presentation

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On some topological upper bounds of the apex trees 1/44

On some topological upper bounds of the apex trees

Sarfraz Ahmad Department of Mathematics, COMSATS University Islamabad, Lahore-Campus Pakistan

Sarfraz Ahmad On some topological upper bounds of the apex trees

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Outline

1

Abstract

2

Introduction and Definitions

3

Main Results Atom-bond connectivity index of k-apex trees Augmented Zagreb index of k-apex trees Geometric-arithmetic index of k-apex trees Inverse sum indeg index of k-apex trees Degree distance index of k-apex trees

4

References

Sarfraz Ahmad On some topological upper bounds of the apex trees

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On some topological upper bounds of the apex trees 3/44 Abstract

Abstract

If a graph G turned out to be a planar graph by removal of a vertex (or a set of vertices) of G, then it is called an apex graph. These graphs play a vital role in the chemical graph theory. On the similar way, a k-apex tree T k

n is a graph which turned out to be a

tree after a removal of k vertices such that k is minimum with this

property. Here |V (T k

n )| = n, the cardinality of the set of vertices.

In this article, we study different topological indices of apex trees. In particular, we provide upper bounds for the geometric-arithmetic index, the atom bond connectivity index, the augmented Zagreb index and the inverse sum index of the apex and k-apex trees. We identity the graphs for which the equalities hold.

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On some topological upper bounds of the apex trees 4/44 Introduction and Definitions

k-apex tree

Let k ≥ 1 be a positive integer. A k-apex tree T k(n) is a graph

with |V (T k(n))| = n and a subset X of V (T kk(n)) satisfying following conditions

Sarfraz Ahmad On some topological upper bounds of the apex trees

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On some topological upper bounds of the apex trees 4/44 Introduction and Definitions

k-apex tree

Let k ≥ 1 be a positive integer. A k-apex tree T k(n) is a graph

with |V (T k(n))| = n and a subset X of V (T kk(n)) satisfying following conditions

T k(n) − X is a tree with |X| = k

Sarfraz Ahmad On some topological upper bounds of the apex trees

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On some topological upper bounds of the apex trees 4/44 Introduction and Definitions

k-apex tree

Let k ≥ 1 be a positive integer. A k-apex tree T k(n) is a graph

with |V (T k(n))| = n and a subset X of V (T kk(n)) satisfying following conditions

T k(n) − X is a tree with |X| = k
for any other subset Y of V (T k(n)) with |y| < k, T k(n) − Y is

not a tree.

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On some topological upper bounds of the apex trees 4/44 Introduction and Definitions

k-apex tree

Let k ≥ 1 be a positive integer. A k-apex tree T k(n) is a graph

with |V (T k(n))| = n and a subset X of V (T kk(n)) satisfying following conditions

T k(n) − X is a tree with |X| = k
for any other subset Y of V (T k(n)) with |y| < k, T k(n) − Y is

not a tree.

The elements of X are called k-apex vertices. If k = 1, T(n) is

called apex tree.

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k-apex tree

Figure: An apex tree on 7 vertices

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k-apex tree

Figure: A 2-apex tree on 5 vertices

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Chemical graph theory

Cheminformatics is new subject which is a combination of

chemistry, mathematics and information science.

Sarfraz Ahmad On some topological upper bounds of the apex trees

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On some topological upper bounds of the apex trees 7/44 Introduction and Definitions

Chemical graph theory

Cheminformatics is new subject which is a combination of

chemistry, mathematics and information science.

Mathematical chemistry is a branch of theoretical chemistry in

which we discuss and predict the chemical structure by using mathematical tools.

Sarfraz Ahmad On some topological upper bounds of the apex trees

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On some topological upper bounds of the apex trees 7/44 Introduction and Definitions

Chemical graph theory

Cheminformatics is new subject which is a combination of

chemistry, mathematics and information science.

Mathematical chemistry is a branch of theoretical chemistry in

which we discuss and predict the chemical structure by using mathematical tools.

Combinatorial Chemistry is a branch of theoretical chemistry in

which we discuss and predict the chemical structure by using combinatorial tools.

Sarfraz Ahmad On some topological upper bounds of the apex trees

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On some topological upper bounds of the apex trees 7/44 Introduction and Definitions

Chemical graph theory

Cheminformatics is new subject which is a combination of

chemistry, mathematics and information science.

Mathematical chemistry is a branch of theoretical chemistry in

which we discuss and predict the chemical structure by using mathematical tools.

Combinatorial Chemistry is a branch of theoretical chemistry in

which we discuss and predict the chemical structure by using combinatorial tools.

Chemical graph theory is branch of Mathematical Chemistry in

which we apply tools from graph theory to model the chemical phenomenon mathematically.

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On some topological upper bounds of the apex trees 8/44 Introduction and Definitions

Topological index

A topological index is a numeric quantity associated with

chemical constitution purporting for correlation of chemical structure with many physico-chemical properties, chemical reactivity or biological activity.

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On some topological upper bounds of the apex trees 8/44 Introduction and Definitions

Topological index

A topological index is a numeric quantity associated with

chemical constitution purporting for correlation of chemical structure with many physico-chemical properties, chemical reactivity or biological activity.

A topological index is a function

Top : Σ − → R where Σ is the set of finite simple graphs and R is the set of real numbers with property that Top(G) = Top(H) if both G and H are isomorphic.

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On some topological upper bounds of the apex trees 9/44 Introduction and Definitions

Wiener Index

The concept of topological index came from work done by Harold Wiener in 1947 while he was working on boiling point of paraffin. He named this index as path number. Later on, path number was renamed as Wiener index and then theory of topological indices started.

Sarfraz Ahmad On some topological upper bounds of the apex trees

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On some topological upper bounds of the apex trees 9/44 Introduction and Definitions

Wiener Index

The concept of topological index came from work done by Harold Wiener in 1947 while he was working on boiling point of paraffin. He named this index as path number. Later on, path number was renamed as Wiener index and then theory of topological indices started. Definition Let G be a graph. Then the Wiener index of G is defined as W (G) = 1

2Σ(u;v)d(u, v) where (u; v) is any ordered pair of vertices

in G and d(u; v) is u − v geodesic.

Sarfraz Ahmad On some topological upper bounds of the apex trees

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On some topological upper bounds of the apex trees 9/44 Introduction and Definitions

Wiener Index

The concept of topological index came from work done by Harold Wiener in 1947 while he was working on boiling point of paraffin. He named this index as path number. Later on, path number was renamed as Wiener index and then theory of topological indices started. Definition Let G be a graph. Then the Wiener index of G is defined as W (G) = 1

2Σ(u;v)d(u, v) where (u; v) is any ordered pair of vertices

in G and d(u; v) is u − v geodesic.

H. Wiener, J. Amer. Chem. Soc., 69(1947); 17 − −20.

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On some topological upper bounds of the apex trees 10/44 Introduction and Definitions

The Zagreb indices

The first Zagreb index M1 was developed by Gutman and Trinajsti` c in 1972. This index is significant in the sense that it has many chemical properties and so it attracted many chemists and mathematicians. Definition The M1 index of G is defined as: M1(G) =

uv∈E(G)

(dG(u) + dG(v)) .

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On some topological upper bounds of the apex trees 11/44 Introduction and Definitions

The Zagreb indices

The first Zagreb coindex M1 was introduced by Doˇ sli´ c. Definition The M1 of G is defined as M1(G): M1(G) =

u,v /

∈E(G)u=v

(dG(u) + dG(v)) .

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The Randi´ c and GA index

Definition The Randi´ c connectivity index of G is given as: R(G) =

uv∈E(G)

1

dG(u)dG(v)

. Thus clearly value of R(G) depends upon the degrees of the vertices connected with edges. Later Vukiˇ cevi´ c and Furtula introduced a topological index named as the geometric-arithmetic index (GA index). It is also about the degrees of the vertices connected with edges defined by: GA(G) =

uv∈E(G)

2

dG(u)dG(v)

dG(u) + dG(v) .

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The atom-bond connectivity index

In 1998 E. Estrada et al. , introduced the atom-bond connectivity index known as the ABC index. The ABC index is very effective topological invariant used to study alkanes energies. The ABC index is some how improvement of the connectivity index χ given by Milan Randi. Definition The ABC index is given by: ABC(G) =

uv∈E(G)
dG(u) + dG(v) − 2

dG(u)dG(v) .

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The augmented Zagreb index

Later Furtula et al. further improved the ABC index and named it as the augmented Zagreb index (AZI). In the study of alkanes and

ctanes energies it was proved that the AZI index performed much

better than the ABC index. Definition The AZI index is defined as: AZI(G) =

uv∈E(G)

( dG(u)dG(v) dG(u) + dG(v) − 2)3.

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The inverse sum indeg index

In [?], to estimate the total surface are of octane isomers a new topological index, known as the inverse sum indeg index (ISI), is used. Definition The ISI has comparatively simple structure and is defined as: ISI(G) =

uv∈E(G)

1

1 dG (u) + 1 dG (v)

=

uv∈E(G)

dG(u)dG(v) dG(u) + dG(v).

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The degree distance index

Dobrynin and Kochetova introduced the degree distance index DD. This index is a weighted version of Wiener index. Definition The degree distance index is defined as: DD(G) =

uv⊂E(G)

dG(u, v)(dG(u) + dG(v)) .

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On some topological upper bounds of the apex trees 17/44 Main Results

In this section we compute upper bounds for the geometric-arithmetic index (GA), the atom-bond connectivity index (ABC), the augmented Zagreb index (AZI), the inverse sum indeg index (ISI) and the degree distance index (DD) of k-apex

trees. Note that T3 is cycle of length 3, so we only consider Tn for

n ≥ 4.

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On some topological upper bounds of the apex trees 18/44 Main Results Atom-bond connectivity index of k-apex trees

Abstract

1

Abstract

2

Introduction and Definitions

3

Main Results Atom-bond connectivity index of k-apex trees Augmented Zagreb index of k-apex trees Geometric-arithmetic index of k-apex trees Inverse sum indeg index of k-apex trees Degree distance index of k-apex trees

4

References

Sarfraz Ahmad On some topological upper bounds of the apex trees

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On some topological upper bounds of the apex trees 19/44 Main Results Atom-bond connectivity index of k-apex trees

Theorem Let G ∈ T(n) be an apex tree on n vertices. If n ≥ 4, then ABC(G) <

n(n − 1)

3

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On some topological upper bounds of the apex trees 20/44 Main Results Atom-bond connectivity index of k-apex trees

Theorem Let G ∈ T 2(n) be 2-apex tree on n vertices. If n ≥ 5, then ABC(G) ≤ √2n − 4 n − 1 + 1 √n − 1 4 √ 3 √n + (n − 4) √ n + 1

+

√ 6 4 (n − 5) +

5

3 and the equality holds if and only if G = K2 + Pn−2.

Sarfraz Ahmad On some topological upper bounds of the apex trees

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On some topological upper bounds of the apex trees 21/44 Main Results Atom-bond connectivity index of k-apex trees

Theorem Let G ∈ T 2(n) be 2-apex tree on n vertices. If n ≥ 5, then ABC(G) ≤ 3(k − 2) √2n − 4 n − 1 + 2k

n + k − 2

(n − 1)(k + 1) + k(n − k − 2)

n + k − 1

(n − 1)(k + 2) + 2

2k + 1

(k + 1)(k + 2) +(n − k − 3) √ 2k + 2 k + 2 . and the equality holds if and only if G = Kk + Pn−k.

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On some topological upper bounds of the apex trees 22/44 Main Results Augmented Zagreb index of k-apex trees

Abstract

1

Abstract

2

Introduction and Definitions

3

Main Results Atom-bond connectivity index of k-apex trees Augmented Zagreb index of k-apex trees Geometric-arithmetic index of k-apex trees Inverse sum indeg index of k-apex trees Degree distance index of k-apex trees

4

References

Sarfraz Ahmad On some topological upper bounds of the apex trees

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On some topological upper bounds of the apex trees 23/44 Main Results Augmented Zagreb index of k-apex trees

Theorem Let G ∈ T(n) be apex tree on n vertices. If n ≥ 4, then AZI(G) ≤ 27(n − 3)(n − 1 n )3 + 729 64 (n − 4) + 32 and the equality holds if and only if G = K1 + Pn−1.

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On some topological upper bounds of the apex trees 24/44 Main Results Augmented Zagreb index of k-apex trees

Theorem Let G ∈ T 2(n) be 2-apex tree on n vertices. If n ≥ 5, then AZI(G) ≤ 1 8 (n − 1)6 (n − 2)3 + 108 n − 1 n

+

512 27 (n − 5) + 128(n − 4) n − 1 n + 1

and the equality holds if and only if G = K2 + Pn−2.

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On some topological upper bounds of the apex trees 25/44 Main Results Augmented Zagreb index of k-apex trees

Theorem Let G ∈ T k(n) be k-apex tree on n vertices. If k ≥ 3, and n ≥ k + 3, then AZI(G) ≤ 3(k − 2)(n − 1)6 8(n − 2)3 + 2k (n − 1)(k + 1) n + k − 2

+

k(n − k − 2) (n − 1)(k + 2) n + k − 1

+ 2

k2 + 3k + 2 2k + 1

+(n − k − 3)(k + 2)6

8(k + 1)3 . and the equality holds if and only if G = Kk + Pn−k.

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On some topological upper bounds of the apex trees 26/44 Main Results Geometric-arithmetic index of k-apex trees

Abstract

1

Abstract

2

Introduction and Definitions

3

Main Results Atom-bond connectivity index of k-apex trees Augmented Zagreb index of k-apex trees Geometric-arithmetic index of k-apex trees Inverse sum indeg index of k-apex trees Degree distance index of k-apex trees

4

References

Sarfraz Ahmad On some topological upper bounds of the apex trees

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On some topological upper bounds of the apex trees 27/44 Main Results Geometric-arithmetic index of k-apex trees

Lemma If G is a simple connected graph with two non-adjacent vertices u and v, then the following inequality holds. GA(G + uv) > GA(G)

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On some topological upper bounds of the apex trees 28/44 Main Results Geometric-arithmetic index of k-apex trees

Theorem Let G ∈ T(n) be apex tree on n vertices. If n ≥ 4, then GA(G) ≤ 2 √ n − 1( 2 √ 2 n + 1 + √ 3(n − 3) n + 2 ) + 5n + 4 √ 6 − 20 5 and the equality holds if and only if G = K1 + Pn−1.

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On some topological upper bounds of the apex trees 29/44 Main Results Geometric-arithmetic index of k-apex trees

Theorem Let G ∈ T 2(n) be 2-apex tree on n vertices. If n ≥ 5, then GA(G) ≤ 1 + 8

3(n − 1)

n + 2 + 4(n − 4)

4(n − 1)

n + 3 + n + 8 7 √ 3 − 5 and equality holds if and only if G = K2 + Pn−2.

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On some topological upper bounds of the apex trees 30/44 Main Results Geometric-arithmetic index of k-apex trees

Theorem Let G ∈ T k(n) be k-apex tree on n vertices. If k ≥ 3, and n ≥ k + 3, then GA(G) ≤ 3(k − 2) + 4k

(n − 1)(k + 1)

n + k + 2k(n − k − 2)

(n − 1)(k + 2)

n + k + 1 + 4 √ k2 + 3k + 2 2k + 3 + n − k − 3 and equality holds if and only if G = Kk + Pn−k.

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On some topological upper bounds of the apex trees 31/44 Main Results Inverse sum indeg index of k-apex trees

Abstract

1

Abstract

2

Introduction and Definitions

3

Main Results Atom-bond connectivity index of k-apex trees Augmented Zagreb index of k-apex trees Geometric-arithmetic index of k-apex trees Inverse sum indeg index of k-apex trees Degree distance index of k-apex trees

4

References

Sarfraz Ahmad On some topological upper bounds of the apex trees

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On some topological upper bounds of the apex trees 32/44 Main Results Inverse sum indeg index of k-apex trees

Theorem Let G ∈ T(n) be apex tree on n vertices. If n ≥ 4, then ISI(G) ≤ 4(1 − 2 n + 1) + 3 2(n − 4) + 3(n + 15 n + 2 − 6) + 12 5 and the equality holds if and only if G = K1 + Pn−1.

Sarfraz Ahmad On some topological upper bounds of the apex trees

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On some topological upper bounds of the apex trees 33/44 Main Results Inverse sum indeg index of k-apex trees

Theorem Let G ∈ T 2(n) be 2-apex tree on n vertices. If n ≥ 5, then ISI(G) ≤ 8(n − 4)(n − 1) n + 3 + 12(n − 1) n + 2 + 5 2n − 99 14 and the equality holds if and only if G = K2 + Pn−2.

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On some topological upper bounds of the apex trees 34/44 Main Results Inverse sum indeg index of k-apex trees

Theorem Let G ∈ T k

n be k-apex tree on n vertices. If k ≥ 3, and n ≥ k + 3,

then ISI(G) ≤ 3 2(nk − k − 2n) − 1 4k + 6 + (n − k − 3)(k + 2) 2 + 2k(n − 1)(k + 1) n + k + k(n − 1)(k + 2)(n − k − 2) n + k + 1 + 9 2 and the equality holds if and only if G = Kk + Pn−k.

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Abstract

1

Abstract

2

Introduction and Definitions

3

Main Results Atom-bond connectivity index of k-apex trees Augmented Zagreb index of k-apex trees Geometric-arithmetic index of k-apex trees Inverse sum indeg index of k-apex trees Degree distance index of k-apex trees

4

References

Sarfraz Ahmad On some topological upper bounds of the apex trees

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Lemma Let G ∈ T(n) with an apex vertex u and maximum DD. Then

1 δ(G) = 2 2 dG(u) = n − 1. Sarfraz Ahmad On some topological upper bounds of the apex trees

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Lemma Let G1 and G2 be graphs with |V (G1)| = n1 and |V (G2)| = n2. Then DD(G1 + G2) = M1(G1) + M1(G2) + 2(M1(G1) + M1(G2)) − (1) = 2n2|E(G1)| − 2n1|E(G2)| + n1n2(3n1 + 3n2 − 4) + (2) = n2

a∈V (G1)

dG1(a) + n1

b∈V (G2)

dG2(b). (3)

Sarfraz Ahmad On some topological upper bounds of the apex trees

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On some topological upper bounds of the apex trees 38/44 Main Results Degree distance index of k-apex trees

Theorem Let G ∈ T(n) be an apex tree. If n ≥ 4, then DD(G) ≤ 4n2 − 10n + 6 + 2M1(Tn−1). with equality holding if and only if G = Sn−1 + K1.

Sarfraz Ahmad On some topological upper bounds of the apex trees

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On some topological upper bounds of the apex trees 39/44 Main Results Degree distance index of k-apex trees

Theorem Let G ∈ T k(n) be a k-apex tree of order n for k ≥ 2. If n ≥ k + 2, then DD(G) ≤ 3nk(n − k − 2) + n(n − 1) + 6k2 + 2M1(T(n − k)) with equality holding if and only if G = Sn−k + Kk.

Sarfraz Ahmad On some topological upper bounds of the apex trees

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Sarfraz Ahmad On some topological upper bounds of the apex trees

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Sarfraz Ahmad On some topological upper bounds of the apex trees

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Sarfraz Ahmad On some topological upper bounds of the apex trees

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cevi´ c, Atom-bond connectivity index of

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Thank You

Sarfraz Ahmad On some topological upper bounds of the apex trees