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The bounds of vertex Padmakar-Ivan index of k-trees

Shaohui Wang

Department of Mathematics and Physics, Texas A&M International University Joint with Zehui Shao, Jiabao Liu and Bing Wei

31st Cumberland Conference University of Central Florida, Orlando, FL

Shaohui Wang The bounds of vertex Padmakar-Ivan index of k-trees

Outline

Introduction

k-trees Padmakar-Ivan Index

Shaohui Wang The bounds of vertex Padmakar-Ivan index of k-trees

Outline

Introduction

k-trees Padmakar-Ivan Index

Our results Some proofs

Shaohui Wang The bounds of vertex Padmakar-Ivan index of k-trees

k-trees

Definition (Beineke and Pippert 1969) The k-tree, denoted by T k

n , for positive integers n, k with n ≥ k,

is defined recursively as follows: The smallest k-tree is the k-clique Kk. If G is a k-tree with n ≥ k vertices and a new vertex v of degree k is added and joined to the vertices of a k-clique in G, then the obtained graph is a k-tree with n + 1 vertices.

Shaohui Wang The bounds of vertex Padmakar-Ivan index of k-trees

k-trees

Definition (Beineke and Pippert 1969) The k-tree, denoted by T k

n , for positive integers n, k with n ≥ k,

is defined recursively as follows: The smallest k-tree is the k-clique Kk. If G is a k-tree with n ≥ k vertices and a new vertex v of degree k is added and joined to the vertices of a k-clique in G, then the obtained graph is a k-tree with n + 1 vertices. Example (Building a 2-tree) 1 2

k-trees

Definition (Beineke and Pippert 1969) The k-tree, denoted by T k

n , for positive integers n, k with n ≥ k,

is defined recursively as follows: The smallest k-tree is the k-clique Kk. If G is a k-tree with n ≥ k vertices and a new vertex v of degree k is added and joined to the vertices of a k-clique in G, then the obtained graph is a k-tree with n + 1 vertices. Example (Building a 2-tree) 1 2 3

k-trees

Definition (Beineke and Pippert 1969) The k-tree, denoted by T k

n , for positive integers n, k with n ≥ k,

is defined recursively as follows: The smallest k-tree is the k-clique Kk. If G is a k-tree with n ≥ k vertices and a new vertex v of degree k is added and joined to the vertices of a k-clique in G, then the obtained graph is a k-tree with n + 1 vertices. Example (Building a 2-tree) 1 2 3 4

k-trees

Definition (Beineke and Pippert 1969) The k-tree, denoted by T k

n , for positive integers n, k with n ≥ k,

is defined recursively as follows: The smallest k-tree is the k-clique Kk. If G is a k-tree with n ≥ k vertices and a new vertex v of degree k is added and joined to the vertices of a k-clique in G, then the obtained graph is a k-tree with n + 1 vertices. Example (Building a 2-tree) 1 2 3 4 5

k-trees

Definition (Beineke and Pippert 1969) The k-tree, denoted by T k

n , for positive integers n, k with n ≥ k,

is defined recursively as follows: The smallest k-tree is the k-clique Kk. If G is a k-tree with n ≥ k vertices and a new vertex v of degree k is added and joined to the vertices of a k-clique in G, then the obtained graph is a k-tree with n + 1 vertices. Example (Building a 2-tree) 1 2 3 4 5 6

k-trees

Definition (Beineke and Pippert 1969) The k-tree, denoted by T k

n , for positive integers n, k with n ≥ k,

is defined recursively as follows: The smallest k-tree is the k-clique Kk. If G is a k-tree with n ≥ k vertices and a new vertex v of degree k is added and joined to the vertices of a k-clique in G, then the obtained graph is a k-tree with n + 1 vertices. Example (Building a 2-tree) 1 2 3 4 5 6 7

Shaohui Wang The bounds of vertex Padmakar-Ivan index of k-trees

k-simplicial vertex

Definition A vertex v ∈ V (T k

n ) is called a k-simplicial vertex if v is a vertex

• f degree k whose neighbors form a k-clique of T k

n .

Shaohui Wang The bounds of vertex Padmakar-Ivan index of k-trees

k-simplicial vertex

Definition A vertex v ∈ V (T k

n ) is called a k-simplicial vertex if v is a vertex

• f degree k whose neighbors form a k-clique of T k

n .

In the following 2-tree, 5, 6, 7 are 2-simplicial vertices. 1 2 3 4 5 6 7

Shaohui Wang The bounds of vertex Padmakar-Ivan index of k-trees

k-simplicial vertex

Let S1(T k

n ) be the set of all simplicial vertices of T k n , for

n ≥ k + 2, and set S1(Kk) = φ, S1(Kk+1) = {v}, where v is any vertex of Kk+1.

Shaohui Wang The bounds of vertex Padmakar-Ivan index of k-trees

k-simplicial vertex

Let S1(T k

n ) be the set of all simplicial vertices of T k n , for

n ≥ k + 2, and set S1(Kk) = φ, S1(Kk+1) = {v}, where v is any vertex of Kk+1. Let G = G0, Gi = Gi−1 − vi, where vi is a simplicial vertex of Gi−1, then {v1, v2, ..., vn} is called a simplicial elimination

• rdering of the n-vertex graph G.

Shaohui Wang The bounds of vertex Padmakar-Ivan index of k-trees

k-path and k-star

The k-path, denoted by Pk

n , for positive integers n, k with

n ≥ k, is defined as follows: Starting with a k-clique G[{v1, v2, ..., vk}]. For i ∈ [k + 1, n], the vertex vi is adjacent to vertices {vi−1, vi−2, ..., vi−k} only.

Shaohui Wang The bounds of vertex Padmakar-Ivan index of k-trees

k-path and k-star

The k-path, denoted by Pk

n , for positive integers n, k with

n ≥ k, is defined as follows: Starting with a k-clique G[{v1, v2, ..., vk}]. For i ∈ [k + 1, n], the vertex vi is adjacent to vertices {vi−1, vi−2, ..., vi−k} only. Example (Building a 2-path) 1 2

k-path and k-star

The k-path, denoted by Pk

n , for positive integers n, k with

n ≥ k, is defined as follows: Starting with a k-clique G[{v1, v2, ..., vk}]. For i ∈ [k + 1, n], the vertex vi is adjacent to vertices {vi−1, vi−2, ..., vi−k} only. Example (Building a 2-path) 1 2 3

k-path and k-star

The k-path, denoted by Pk

n , for positive integers n, k with

n ≥ k, is defined as follows: Starting with a k-clique G[{v1, v2, ..., vk}]. For i ∈ [k + 1, n], the vertex vi is adjacent to vertices {vi−1, vi−2, ..., vi−k} only. Example (Building a 2-path) 1 2 3 4

k-path and k-star

The k-path, denoted by Pk

n , for positive integers n, k with

n ≥ k, is defined as follows: Starting with a k-clique G[{v1, v2, ..., vk}]. For i ∈ [k + 1, n], the vertex vi is adjacent to vertices {vi−1, vi−2, ..., vi−k} only. Example (Building a 2-path) 1 2 3 4 5

k-path and k-star

The k-path, denoted by Pk

n , for positive integers n, k with

n ≥ k, is defined as follows: Starting with a k-clique G[{v1, v2, ..., vk}]. For i ∈ [k + 1, n], the vertex vi is adjacent to vertices {vi−1, vi−2, ..., vi−k} only. Example (Building a 2-path) 1 2 3 4 5 6

k-path and k-star

The k-path, denoted by Pk

n , for positive integers n, k with

n ≥ k, is defined as follows: Starting with a k-clique G[{v1, v2, ..., vk}]. For i ∈ [k + 1, n], the vertex vi is adjacent to vertices {vi−1, vi−2, ..., vi−k} only. Example (Building a 2-path) 1 2 3 4 5 6 7

Shaohui Wang The bounds of vertex Padmakar-Ivan index of k-trees

The k-star, denoted by Sk

n , for positive integers n, k with

n ≥ k, is defined as follows: Starting with a k-clique G[{v1, v2, ..., vk}] and an independent set S with |S| = n − k. For i ∈ [k + 1, n], the vertex vi is adjacent to vertices {v1, v2, ..., vk} only.

Shaohui Wang The bounds of vertex Padmakar-Ivan index of k-trees

The k-star, denoted by Sk

n , for positive integers n, k with

n ≥ k, is defined as follows: Starting with a k-clique G[{v1, v2, ..., vk}] and an independent set S with |S| = n − k. For i ∈ [k + 1, n], the vertex vi is adjacent to vertices {v1, v2, ..., vk} only. Example (Building a 2-star) 1 2

The k-star, denoted by Sk

n , for positive integers n, k with

n ≥ k, is defined as follows: Starting with a k-clique G[{v1, v2, ..., vk}] and an independent set S with |S| = n − k. For i ∈ [k + 1, n], the vertex vi is adjacent to vertices {v1, v2, ..., vk} only. Example (Building a 2-star) 1 2 3

The k-star, denoted by Sk

n , for positive integers n, k with

n ≥ k, is defined as follows: Starting with a k-clique G[{v1, v2, ..., vk}] and an independent set S with |S| = n − k. For i ∈ [k + 1, n], the vertex vi is adjacent to vertices {v1, v2, ..., vk} only. Example (Building a 2-star) 1 2 3 4

The k-star, denoted by Sk

n , for positive integers n, k with

n ≥ k, is defined as follows: Starting with a k-clique G[{v1, v2, ..., vk}] and an independent set S with |S| = n − k. For i ∈ [k + 1, n], the vertex vi is adjacent to vertices {v1, v2, ..., vk} only. Example (Building a 2-star) 1 2 3 4 5

The k-star, denoted by Sk

n , for positive integers n, k with

n ≥ k, is defined as follows: Starting with a k-clique G[{v1, v2, ..., vk}] and an independent set S with |S| = n − k. For i ∈ [k + 1, n], the vertex vi is adjacent to vertices {v1, v2, ..., vk} only. Example (Building a 2-star) 1 2 3 4 5 6

The k-star, denoted by Sk

n , for positive integers n, k with

n ≥ k, is defined as follows: Starting with a k-clique G[{v1, v2, ..., vk}] and an independent set S with |S| = n − k. For i ∈ [k + 1, n], the vertex vi is adjacent to vertices {v1, v2, ..., vk} only. Example (Building a 2-star) 1 2 3 4 5 6 7

Shaohui Wang The bounds of vertex Padmakar-Ivan index of k-trees

Zagreb Indices

Definition (Gutman and Trinajsti´ c 1972) The first and second Zagreb indices of the graph G = (V , E) are defined as M1 =

• v∈V (G)

d(v)2; M2 =

• uv∈E(G)

d(u)d(v).

Shaohui Wang The bounds of vertex Padmakar-Ivan index of k-trees

Zagreb Indices

Definition (Gutman and Trinajsti´ c 1972) The first and second Zagreb indices of the graph G = (V , E) are defined as M1 =

• v∈V (G)

d(v)2; M2 =

• uv∈E(G)

d(u)d(v). Definition (Todeschini etc 2010; Wang and Wei 2015) The first(generalized) and second Multiplicative Zagreb indices

• f the graph G = (V , E) are defined as
• 1,c

(G) =

• v∈V (G)

d(v)c, c ≥ 1;

• 2

(G) =

• uv∈E(G)

d(u)d(v).

Shaohui Wang The bounds of vertex Padmakar-Ivan index of k-trees

Theorem (Das and Gutman 2004) Let T be any tree on n vertices, then M1(Pn) ≤ M1(T) ≤ M1(Sn), M2(Pn) ≤ M2(T) ≤ M2(Sn).

Shaohui Wang The bounds of vertex Padmakar-Ivan index of k-trees

Theorem (Das and Gutman 2004) Let T be any tree on n vertices, then M1(Pn) ≤ M1(T) ≤ M1(Sn), M2(Pn) ≤ M2(T) ≤ M2(Sn). Theorem (Estes and Wei 2012) Let T k

n be any k-tree on n vertices, then

M1(Pk

n ) ≤ M1(T k n ) ≤ M1(Sk n ),

M2(Pk

n ) ≤ M2(T k n ) ≤ M2(Sk n ).

Shaohui Wang The bounds of vertex Padmakar-Ivan index of k-trees

Theorem (Gutman 2011) Let n ≥ 5 and Tn be any tree with n vertices, then

• 1

(Sn) ≤

• 1

(Tn) ≤

• 1

(Pn),

• 2

(Pn) ≤

• 2

(Tn) ≤

• 2

(Sn).

Shaohui Wang The bounds of vertex Padmakar-Ivan index of k-trees

Theorem (Gutman 2011) Let n ≥ 5 and Tn be any tree with n vertices, then

• 1

(Sn) ≤

• 1

(Tn) ≤

• 1

(Pn),

• 2

(Pn) ≤

• 2

(Tn) ≤

• 2

(Sn). Theorem (Wang and Wei 2015) Let T k

n be a k-tree on n ≥ k vertices, then

• 1,c

(Sk

n ) ≤

• 1,c

(T k

n ) ≤

• 1,c

(Pk

n ),

• 2

(Pk

n ) ≤

• 2

(T k

n ) ≤

• 2

(Sk

n ).

Shaohui Wang The bounds of vertex Padmakar-Ivan index of k-trees

Padmakar-Ivan Indices

Definition (Winener 1947) The Wiener Index of the graph G = (V , E) are defined as W (G) =

• x,y∈V (G)

d(x, y).

Shaohui Wang The bounds of vertex Padmakar-Ivan index of k-trees

Padmakar-Ivan Indices

For any xy ∈ E(G), let nxy(x) be the number of vertices w ∈ V (G) such that d(x, w) < d(y, w). Definition (Gutman 1994) The Szeged index of the graph G = (V , E) are defined as Sz(G) =

• xy∈E(G)

nxy(x)nxy(y).

Shaohui Wang The bounds of vertex Padmakar-Ivan index of k-trees

Padmakar-Ivan Indices

For any xy ∈ E(G), let nxy(x) be the number of vertices w ∈ V (G) such that d(x, w) < d(y, w). Definition (Gutman 1994) The Szeged index of the graph G = (V , E) are defined as Sz(G) =

• xy∈E(G)

nxy(x)nxy(y). Definition (Khadikar 2000) The Padmakar-Ivan index of the graph G = (V , E) is defined as PI(G) =

• xy∈E(G)

[nxy(x) + nxy(y)]

Shaohui Wang The bounds of vertex Padmakar-Ivan index of k-trees

Known Results

Ashrafi and Loghman determined exact PI index of zig-zag polyhex nanotubes in 2006.

Shaohui Wang The bounds of vertex Padmakar-Ivan index of k-trees

Known Results

Ashrafi and Loghman determined exact PI index of zig-zag polyhex nanotubes in 2006. Klavzar provided the PI index: PI-partitions and Cartesian product graphs in 2007.

Shaohui Wang The bounds of vertex Padmakar-Ivan index of k-trees

Known Results

Ashrafi and Loghman determined exact PI index of zig-zag polyhex nanotubes in 2006. Klavzar provided the PI index: PI-partitions and Cartesian product graphs in 2007. Pattabiraman and Kandan discovered the product and join of two graphs for weighted PI indices in 2016.

Shaohui Wang The bounds of vertex Padmakar-Ivan index of k-trees

Known Results

Ashrafi and Loghman determined exact PI index of zig-zag polyhex nanotubes in 2006. Klavzar provided the PI index: PI-partitions and Cartesian product graphs in 2007. Pattabiraman and Kandan discovered the product and join of two graphs for weighted PI indices in 2016. Illic solved an open problem that the complete bipartite graph by adding one edge connecting two vertices from the class of size is the unique graph with the second-maximal value of PI index in 2010.

Shaohui Wang The bounds of vertex Padmakar-Ivan index of k-trees

Known Results

Ashrafi and Loghman determined exact PI index of zig-zag polyhex nanotubes in 2006. Klavzar provided the PI index: PI-partitions and Cartesian product graphs in 2007. Pattabiraman and Kandan discovered the product and join of two graphs for weighted PI indices in 2016. Illic solved an open problem that the complete bipartite graph by adding one edge connecting two vertices from the class of size is the unique graph with the second-maximal value of PI index in 2010. Yarahmadi et al. gave the first and second extremal bipartite graphs with respect to edge PI index in 2011.

Shaohui Wang The bounds of vertex Padmakar-Ivan index of k-trees

Known Results

Ashrafi and Loghman determined exact PI index of zig-zag polyhex nanotubes in 2006. Klavzar provided the PI index: PI-partitions and Cartesian product graphs in 2007. Pattabiraman and Kandan discovered the product and join of two graphs for weighted PI indices in 2016. Illic solved an open problem that the complete bipartite graph by adding one edge connecting two vertices from the class of size is the unique graph with the second-maximal value of PI index in 2010. Yarahmadi et al. gave the first and second extremal bipartite graphs with respect to edge PI index in 2011. Vukicevic and Stevanovic characterized bicyclic graphs with extremal values of edge PI index in 2013.

Shaohui Wang The bounds of vertex Padmakar-Ivan index of k-trees

Known Results

Ashrafi and Loghman determined exact PI index of zig-zag polyhex nanotubes in 2006. Klavzar provided the PI index: PI-partitions and Cartesian product graphs in 2007. Pattabiraman and Kandan discovered the product and join of two graphs for weighted PI indices in 2016. Illic solved an open problem that the complete bipartite graph by adding one edge connecting two vertices from the class of size is the unique graph with the second-maximal value of PI index in 2010. Yarahmadi et al. gave the first and second extremal bipartite graphs with respect to edge PI index in 2011. Vukicevic and Stevanovic characterized bicyclic graphs with extremal values of edge PI index in 2013. Klavar studied the difference between the Szeged index(PI index) and the Wiener index of cacti in 2018.

Shaohui Wang The bounds of vertex Padmakar-Ivan index of k-trees

Known Results

Let B be any bipartite graph on n vertices and m edges, then ”every edge of B has the PI-value as n − 2” and PI(B) = (n − 2)m.

Shaohui Wang The bounds of vertex Padmakar-Ivan index of k-trees

Known Results

Let B be any bipartite graph on n vertices and m edges, then ”every edge of B has the PI-value as n − 2” and PI(B) = (n − 2)m. In particular, if B is a tree, PI(B) = (n-1)(n-2).

Shaohui Wang The bounds of vertex Padmakar-Ivan index of k-trees

Known Results

Let B be any bipartite graph on n vertices and m edges, then ”every edge of B has the PI-value as n − 2” and PI(B) = (n − 2)m. In particular, if B is a tree, PI(B) = (n-1)(n-2). Question: For k-trees, do the PI values are the same?

Shaohui Wang The bounds of vertex Padmakar-Ivan index of k-trees

Main results

Not all k-trees have the same PI values. Theorem (Wang, Shao, Liu, Wei 2019) For any k-star Sk

n and k-path Pk n with n = kp + s vertices, where

2 ≤ s ≤ k + 1 and p ≥ 1, we have (i)PI(Sk

n ) = k(n − k)(n − k − 1),

(ii)PI(Pk

n ) = k(k+1)(p−1)(3kp+6s−2k−4) 6

+ (s−1)s(3k−s+2)

3

.

Shaohui Wang The bounds of vertex Padmakar-Ivan index of k-trees

Main results

Not all k-trees have the same PI values. Theorem (Wang, Shao, Liu, Wei 2019) For any k-star Sk

n and k-path Pk n with n = kp + s vertices, where

2 ≤ s ≤ k + 1 and p ≥ 1, we have (i)PI(Sk

n ) = k(n − k)(n − k − 1),

(ii)PI(Pk

n ) = k(k+1)(p−1)(3kp+6s−2k−4) 6

+ (s−1)s(3k−s+2)

3

. The k-star arrives the maximal PI values among all k-trees. Theorem (Wang, Shao, Liu, Wei 2019) Let T k

n be any k-tree on n ≥ k ≥ 1. Then PI(T k n ) ≤ PI(Sk n ).

Shaohui Wang The bounds of vertex Padmakar-Ivan index of k-trees

Main results

Not all k-trees have the same PI values. Theorem (Wang, Shao, Liu, Wei 2019) For any k-star Sk

n and k-path Pk n with n = kp + s vertices, where

2 ≤ s ≤ k + 1 and p ≥ 1, we have (i)PI(Sk

n ) = k(n − k)(n − k − 1),

(ii)PI(Pk

n ) = k(k+1)(p−1)(3kp+6s−2k−4) 6

+ (s−1)s(3k−s+2)

3

. The k-star arrives the maximal PI values among all k-trees. Theorem (Wang, Shao, Liu, Wei 2019) Let T k

n be any k-tree on n ≥ k ≥ 1. Then PI(T k n ) ≤ PI(Sk n ).

How about the minimal PI values among all k-trees?

Shaohui Wang The bounds of vertex Padmakar-Ivan index of k-trees

k-Spiral

Definition The k-spiral, denoted by T k∗

n,c with c ∈ [1, k − 1], is defined as

Pk−c

n−c + Kc, that is, V (T k∗ n,c) = {v1, v2, ..., vn} and

E(T k∗

n,c) = E(Pk−1 n−1 ) ∪ E(Kc) ∪ {v1vl, v2vl, ..., vn−cvl}, for

l ∈ [n − c + 1, n].

Shaohui Wang The bounds of vertex Padmakar-Ivan index of k-trees

k-Spiral

Definition The k-spiral, denoted by T k∗

n,c with c ∈ [1, k − 1], is defined as

Pk−c

n−c + Kc, that is, V (T k∗ n,c) = {v1, v2, ..., vn} and

E(T k∗

n,c) = E(Pk−1 n−1 ) ∪ E(Kc) ∪ {v1vl, v2vl, ..., vn−cvl}, for

l ∈ [n − c + 1, n]. Example (Building a 3-spiral with c = 2, say T 3∗

7,2)

1 2 3

k-Spiral

Definition The k-spiral, denoted by T k∗

n,c with c ∈ [1, k − 1], is defined as

Pk−c

n−c + Kc, that is, V (T k∗ n,c) = {v1, v2, ..., vn} and

E(T k∗

n,c) = E(Pk−1 n−1 ) ∪ E(Kc) ∪ {v1vl, v2vl, ..., vn−cvl}, for

l ∈ [n − c + 1, n]. Example (Building a 3-spiral with c = 2, say T 3∗

7,2)

1 2 3 4

k-Spiral

Definition The k-spiral, denoted by T k∗

n,c with c ∈ [1, k − 1], is defined as

Pk−c

n−c + Kc, that is, V (T k∗ n,c) = {v1, v2, ..., vn} and

E(T k∗

n,c) = E(Pk−1 n−1 ) ∪ E(Kc) ∪ {v1vl, v2vl, ..., vn−cvl}, for

l ∈ [n − c + 1, n]. Example (Building a 3-spiral with c = 2, say T 3∗

7,2)

1 2 3 4 5

k-Spiral

Definition The k-spiral, denoted by T k∗

n,c with c ∈ [1, k − 1], is defined as

Pk−c

n−c + Kc, that is, V (T k∗ n,c) = {v1, v2, ..., vn} and

E(T k∗

n,c) = E(Pk−1 n−1 ) ∪ E(Kc) ∪ {v1vl, v2vl, ..., vn−cvl}, for

l ∈ [n − c + 1, n]. Example (Building a 3-spiral with c = 2, say T 3∗

7,2)

1 2 3 4 5 6

k-Spiral

Definition The k-spiral, denoted by T k∗

n,c with c ∈ [1, k − 1], is defined as

Pk−c

n−c + Kc, that is, V (T k∗ n,c) = {v1, v2, ..., vn} and

E(T k∗

n,c) = E(Pk−1 n−1 ) ∪ E(Kc) ∪ {v1vl, v2vl, ..., vn−cvl}, for

l ∈ [n − c + 1, n]. Example (Building a 3-spiral with c = 2, say T 3∗

7,2)

1 2 3 4 5 6 7

Shaohui Wang The bounds of vertex Padmakar-Ivan index of k-trees

Main results

The partial lower bounds are deduced. Theorem (Wang, Shao, Liu, Wei 2019) For any k-spiral T k∗

n,c with n ≥ k ≥ 1, we have

(i) PI(Pk

n ) ≥ PI(T k∗ n,c) if c ∈ [1, k+1 2 ),

(ii) PI(Pk

n ) ≤ PI(T k∗ n,c) if c ∈ [ k+1 2 , k − 1].

Shaohui Wang The bounds of vertex Padmakar-Ivan index of k-trees

Main results

The partial lower bounds are deduced. Theorem (Wang, Shao, Liu, Wei 2019) For any k-spiral T k∗

n,c with n ≥ k ≥ 1, we have

(i) PI(Pk

n ) ≥ PI(T k∗ n,c) if c ∈ [1, k+1 2 ),

(ii) PI(Pk

n ) ≤ PI(T k∗ n,c) if c ∈ [ k+1 2 , k − 1].

The PI values of a k-spiral: Theorem (Wang, Shao, Liu, Wei 2019) For any k-spiral T k∗

n,c with n ≥ k vertices, where c ∈ [1, k − 1],

PI =     

(n−k)(n−k−1)(4k−n+2) 3

if n ∈ [k, 2k − c],

3c(n−2k+c−1)(n−2k+c)+(k−c)(2c2+3nc−4kc+3kn−4k2−6k+3n−2) 3

if n ≥ 2k − c + 1.

Shaohui Wang The bounds of vertex Padmakar-Ivan index of k-trees

A tool

Let G ′ = G ∪ {u} be a k-tree obtained by adding a new vertex u to G. For any v1, v2 ∈ V (G), let d(v1, v2) be the distance between v1 and v2 in G, d′(v1, v2) be the distance between v1 and v2 in G ′. Build a fuction as follows: f : {w ∈ V (G ′), xy ∈ E(G)} to {1, 0} as follows: f (w, xy) =        if w = u and d′(x, w) = d′(y, w), if w ∈ V (G) and d(x, w) − d′(x, w) = d(y, w) − d′(y, w), 1 if Otherwise.

Shaohui Wang The bounds of vertex Padmakar-Ivan index of k-trees

Some Remarks

The PI-values of k-paths do not attain the minimal values for k-trees and not all of k-spirals are strictly less than any PI-value of other type of k-trees.

Shaohui Wang The bounds of vertex Padmakar-Ivan index of k-trees

Some Remarks

The PI-values of k-paths do not attain the minimal values for k-trees and not all of k-spirals are strictly less than any PI-value of other type of k-trees. The two natural problems are that what is the minimum PI-value for k-trees and which type of k-trees will achieve the minimum PI-value?

Shaohui Wang The bounds of vertex Padmakar-Ivan index of k-trees

Thank you

Shaohui Wang The bounds of vertex Padmakar-Ivan index of k-trees