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 K k . 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 K k . 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 K k . 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 K k . 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 4 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 K k . 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 4 2 3 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 K k . 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 4 6 2 3 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 K k . 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 4 7 6 2 3 5 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 of 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 of 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 4 7 6 2 3 5 Shaohui Wang The bounds of vertex Padmakar-Ivan index of k -trees
k -simplicial vertex Let S 1 ( T k n ) be the set of all simplicial vertices of T k n , for n ≥ k + 2, and set S 1 ( K k ) = φ, S 1 ( K k +1 ) = { v } , where v is any vertex of K k +1 . Shaohui Wang The bounds of vertex Padmakar-Ivan index of k -trees
k -simplicial vertex Let S 1 ( T k n ) be the set of all simplicial vertices of T k n , for n ≥ k + 2, and set S 1 ( K k ) = φ, S 1 ( K k +1 ) = { v } , where v is any vertex of K k +1 . Let G = G 0 , G i = G i − 1 − v i , where v i is a simplicial vertex of G i − 1 , then { v 1 , v 2 , ..., v n } is called a simplicial elimination ordering 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 P k n , for positive integers n , k with n ≥ k , is defined as follows: Starting with a k -clique G [ { v 1 , v 2 , ..., v k } ]. For i ∈ [ k + 1 , n ], the vertex v i is adjacent to vertices { v i − 1 , v i − 2 , ..., v i − k } only. Shaohui Wang The bounds of vertex Padmakar-Ivan index of k -trees
k -path and k -star The k -path , denoted by P k n , for positive integers n , k with n ≥ k , is defined as follows: Starting with a k -clique G [ { v 1 , v 2 , ..., v k } ]. For i ∈ [ k + 1 , n ], the vertex v i is adjacent to vertices { v i − 1 , v i − 2 , ..., v i − k } only. Example (Building a 2-path) 1 2
k -path and k -star The k -path , denoted by P k n , for positive integers n , k with n ≥ k , is defined as follows: Starting with a k -clique G [ { v 1 , v 2 , ..., v k } ]. For i ∈ [ k + 1 , n ], the vertex v i is adjacent to vertices { v i − 1 , v i − 2 , ..., v i − k } only. Example (Building a 2-path) 1 2 3
k -path and k -star The k -path , denoted by P k n , for positive integers n , k with n ≥ k , is defined as follows: Starting with a k -clique G [ { v 1 , v 2 , ..., v k } ]. For i ∈ [ k + 1 , n ], the vertex v i is adjacent to vertices { v i − 1 , v i − 2 , ..., v i − k } only. Example (Building a 2-path) 1 4 2 3
k -path and k -star The k -path , denoted by P k n , for positive integers n , k with n ≥ k , is defined as follows: Starting with a k -clique G [ { v 1 , v 2 , ..., v k } ]. For i ∈ [ k + 1 , n ], the vertex v i is adjacent to vertices { v i − 1 , v i − 2 , ..., v i − k } only. Example (Building a 2-path) 1 4 2 3 5
k -path and k -star The k -path , denoted by P k n , for positive integers n , k with n ≥ k , is defined as follows: Starting with a k -clique G [ { v 1 , v 2 , ..., v k } ]. For i ∈ [ k + 1 , n ], the vertex v i is adjacent to vertices { v i − 1 , v i − 2 , ..., v i − k } only. Example (Building a 2-path) 1 4 6 2 3 5
k -path and k -star The k -path , denoted by P k n , for positive integers n , k with n ≥ k , is defined as follows: Starting with a k -clique G [ { v 1 , v 2 , ..., v k } ]. For i ∈ [ k + 1 , n ], the vertex v i is adjacent to vertices { v i − 1 , v i − 2 , ..., v i − k } only. Example (Building a 2-path) 1 4 6 7 2 3 5 Shaohui Wang The bounds of vertex Padmakar-Ivan index of k -trees
The k -star , denoted by S k n , for positive integers n , k with n ≥ k , is defined as follows: Starting with a k -clique G [ { v 1 , v 2 , ..., v k } ] and an independent set S with | S | = n − k . For i ∈ [ k + 1 , n ], the vertex v i is adjacent to vertices { v 1 , v 2 , ..., v k } only. Shaohui Wang The bounds of vertex Padmakar-Ivan index of k -trees
The k -star , denoted by S k n , for positive integers n , k with n ≥ k , is defined as follows: Starting with a k -clique G [ { v 1 , v 2 , ..., v k } ] and an independent set S with | S | = n − k . For i ∈ [ k + 1 , n ], the vertex v i is adjacent to vertices { v 1 , v 2 , ..., v k } only. Example (Building a 2-star) 1 2
The k -star , denoted by S k n , for positive integers n , k with n ≥ k , is defined as follows: Starting with a k -clique G [ { v 1 , v 2 , ..., v k } ] and an independent set S with | S | = n − k . For i ∈ [ k + 1 , n ], the vertex v i is adjacent to vertices { v 1 , v 2 , ..., v k } only. Example (Building a 2-star) 3 1 2
The k -star , denoted by S k n , for positive integers n , k with n ≥ k , is defined as follows: Starting with a k -clique G [ { v 1 , v 2 , ..., v k } ] and an independent set S with | S | = n − k . For i ∈ [ k + 1 , n ], the vertex v i is adjacent to vertices { v 1 , v 2 , ..., v k } only. Example (Building a 2-star) 3 1 2 4
The k -star , denoted by S k n , for positive integers n , k with n ≥ k , is defined as follows: Starting with a k -clique G [ { v 1 , v 2 , ..., v k } ] and an independent set S with | S | = n − k . For i ∈ [ k + 1 , n ], the vertex v i is adjacent to vertices { v 1 , v 2 , ..., v k } only. Example (Building a 2-star) 5 3 1 2 4
The k -star , denoted by S k n , for positive integers n , k with n ≥ k , is defined as follows: Starting with a k -clique G [ { v 1 , v 2 , ..., v k } ] and an independent set S with | S | = n − k . For i ∈ [ k + 1 , n ], the vertex v i is adjacent to vertices { v 1 , v 2 , ..., v k } only. Example (Building a 2-star) 5 3 6 1 2 4
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