The Subtree Polynomial Lucas Mol Joint work with Jason Brown - PowerPoint PPT Presentation

I NTRODUCTION C OUNTING S UBTREES C OMPLEX S UBTREE R OOTS R EAL S UBTREE R OOTS O PEN P ROBLEMS The Subtree Polynomial Lucas Mol Joint work with Jason Brown (Dalhousie University) CanaDAM 2019 – Graph Polynomials Minisymposium

I NTRODUCTION C OUNTING S UBTREES C OMPLEX S UBTREE R OOTS R EAL S UBTREE R OOTS O PEN P ROBLEMS P LAN I NTRODUCTION C OUNTING S UBTREES C OMPLEX S UBTREE R OOTS R EAL S UBTREE R OOTS O PEN P ROBLEMS

I NTRODUCTION C OUNTING S UBTREES C OMPLEX S UBTREE R OOTS R EAL S UBTREE R OOTS O PEN P ROBLEMS P LAN I NTRODUCTION C OUNTING S UBTREES C OMPLEX S UBTREE R OOTS R EAL S UBTREE R OOTS O PEN P ROBLEMS

I NTRODUCTION C OUNTING S UBTREES C OMPLEX S UBTREE R OOTS R EAL S UBTREE R OOTS O PEN P ROBLEMS T HE S UBTREE P OLYNOMIAL Let T be a tree (or forest), and let S be the collection of all subtrees (connected subgraphs) of T .

I NTRODUCTION C OUNTING S UBTREES C OMPLEX S UBTREE R OOTS R EAL S UBTREE R OOTS O PEN P ROBLEMS T HE S UBTREE P OLYNOMIAL Let T be a tree (or forest), and let S be the collection of all subtrees (connected subgraphs) of T . ◮ Define the subtree polynomial of T by n � x | V ( S ) | . Φ T ( x ) = S ∈S

I NTRODUCTION C OUNTING S UBTREES C OMPLEX S UBTREE R OOTS R EAL S UBTREE R OOTS O PEN P ROBLEMS T HE S UBTREE P OLYNOMIAL Let T be a tree (or forest), and let S be the collection of all subtrees (connected subgraphs) of T . ◮ Define the subtree polynomial of T by n � x | V ( S ) | . Φ T ( x ) = S ∈S ◮ Alternatively, n � s k ( T ) x k , Φ T ( x ) = k = 1 where s k ( T ) is the number of subtrees of T of order k .

I NTRODUCTION C OUNTING S UBTREES C OMPLEX S UBTREE R OOTS R EAL S UBTREE R OOTS O PEN P ROBLEMS A N E XAMPLE Consider the following tree T : Φ T ( x ) =

I NTRODUCTION C OUNTING S UBTREES C OMPLEX S UBTREE R OOTS R EAL S UBTREE R OOTS O PEN P ROBLEMS A N E XAMPLE Consider the following tree T : Φ T ( x ) = 5 x

I NTRODUCTION C OUNTING S UBTREES C OMPLEX S UBTREE R OOTS R EAL S UBTREE R OOTS O PEN P ROBLEMS A N E XAMPLE Consider the following tree T : Φ T ( x ) = 5 x + 4 x 2

I NTRODUCTION C OUNTING S UBTREES C OMPLEX S UBTREE R OOTS R EAL S UBTREE R OOTS O PEN P ROBLEMS A N E XAMPLE Consider the following tree T : Φ T ( x ) = 5 x + 4 x 2 + 4 x 3

I NTRODUCTION C OUNTING S UBTREES C OMPLEX S UBTREE R OOTS R EAL S UBTREE R OOTS O PEN P ROBLEMS A N E XAMPLE Consider the following tree T : Φ T ( x ) = 5 x + 4 x 2 + 4 x 3 + 3 x 4

I NTRODUCTION C OUNTING S UBTREES C OMPLEX S UBTREE R OOTS R EAL S UBTREE R OOTS O PEN P ROBLEMS A N E XAMPLE Consider the following tree T : Φ T ( x ) = 5 x + 4 x 2 + 4 x 3 + 3 x 4 + x 5

I NTRODUCTION C OUNTING S UBTREES C OMPLEX S UBTREE R OOTS R EAL S UBTREE R OOTS O PEN P ROBLEMS G RAPH P ARAMETERS E NCODED

I NTRODUCTION C OUNTING S UBTREES C OMPLEX S UBTREE R OOTS R EAL S UBTREE R OOTS O PEN P ROBLEMS G RAPH P ARAMETERS E NCODED ◮ Φ T ( 1 ) is the number of subtrees of T .

I NTRODUCTION C OUNTING S UBTREES C OMPLEX S UBTREE R OOTS R EAL S UBTREE R OOTS O PEN P ROBLEMS G RAPH P ARAMETERS E NCODED ◮ Φ T ( 1 ) is the number of subtrees of T . ◮ An important topological index, inversely correlated to the Wiener index .

I NTRODUCTION C OUNTING S UBTREES C OMPLEX S UBTREE R OOTS R EAL S UBTREE R OOTS O PEN P ROBLEMS G RAPH P ARAMETERS E NCODED ◮ Φ T ( 1 ) is the number of subtrees of T . ◮ An important topological index, inversely correlated to the Wiener index . ◮ The quantity M T = Φ ′ T ( 1 ) Φ T ( 1 ) is the mean subtree order of T .

I NTRODUCTION C OUNTING S UBTREES C OMPLEX S UBTREE R OOTS R EAL S UBTREE R OOTS O PEN P ROBLEMS G RAPH P ARAMETERS E NCODED ◮ Φ T ( 1 ) is the number of subtrees of T . ◮ An important topological index, inversely correlated to the Wiener index . ◮ The quantity M T = Φ ′ T ( 1 ) Φ T ( 1 ) is the mean subtree order of T . ◮ First studied by Jamison.

I NTRODUCTION C OUNTING S UBTREES C OMPLEX S UBTREE R OOTS R EAL S UBTREE R OOTS O PEN P ROBLEMS G RAPH P ARAMETERS E NCODED ◮ Φ T ( 1 ) is the number of subtrees of T . ◮ An important topological index, inversely correlated to the Wiener index . ◮ The quantity M T = Φ ′ T ( 1 ) Φ T ( 1 ) is the mean subtree order of T . ◮ First studied by Jamison. ◮ Less obvious: − Φ T ( − 1 ) is the independence number of T (Jamsion, 1987).

I NTRODUCTION C OUNTING S UBTREES C OMPLEX S UBTREE R OOTS R EAL S UBTREE R OOTS O PEN P ROBLEMS T HE LOCAL SUBTREE POLYNOMIAL

I NTRODUCTION C OUNTING S UBTREES C OMPLEX S UBTREE R OOTS R EAL S UBTREE R OOTS O PEN P ROBLEMS T HE LOCAL SUBTREE POLYNOMIAL Let T be a tree with vertex v , and let S v be the collection of subtrees of T containing v .

I NTRODUCTION C OUNTING S UBTREES C OMPLEX S UBTREE R OOTS R EAL S UBTREE R OOTS O PEN P ROBLEMS T HE LOCAL SUBTREE POLYNOMIAL Let T be a tree with vertex v , and let S v be the collection of subtrees of T containing v . ◮ The local subtree polynomial of T at v is given by � x | V ( S ) | . Φ T , v ( x ) = S ∈S v

I NTRODUCTION C OUNTING S UBTREES C OMPLEX S UBTREE R OOTS R EAL S UBTREE R OOTS O PEN P ROBLEMS T HE LOCAL SUBTREE POLYNOMIAL Let T be a tree with vertex v , and let S v be the collection of subtrees of T containing v . ◮ The local subtree polynomial of T at v is given by � x | V ( S ) | . Φ T , v ( x ) = S ∈S v Evidently, we have: Φ T ( x ) = Φ T , v ( x ) + Φ T − v ( x ) .

I NTRODUCTION C OUNTING S UBTREES C OMPLEX S UBTREE R OOTS R EAL S UBTREE R OOTS O PEN P ROBLEMS A N E XAMPLE Consider the following tree T with vertex v : v Φ T , v ( x ) =

I NTRODUCTION C OUNTING S UBTREES C OMPLEX S UBTREE R OOTS R EAL S UBTREE R OOTS O PEN P ROBLEMS A N E XAMPLE Consider the following tree T with vertex v : v Φ T , v ( x ) = x

I NTRODUCTION C OUNTING S UBTREES C OMPLEX S UBTREE R OOTS R EAL S UBTREE R OOTS O PEN P ROBLEMS A N E XAMPLE Consider the following tree T with vertex v : v x + x 2 Φ T , v ( x ) =

I NTRODUCTION C OUNTING S UBTREES C OMPLEX S UBTREE R OOTS R EAL S UBTREE R OOTS O PEN P ROBLEMS A N E XAMPLE Consider the following tree T with vertex v : v x + x 2 + 2 x 3 Φ T , v ( x ) =

I NTRODUCTION C OUNTING S UBTREES C OMPLEX S UBTREE R OOTS R EAL S UBTREE R OOTS O PEN P ROBLEMS A N E XAMPLE Consider the following tree T with vertex v : v x + x 2 + 2 x 3 + 2 x 4 Φ T , v ( x ) =

I NTRODUCTION C OUNTING S UBTREES C OMPLEX S UBTREE R OOTS R EAL S UBTREE R OOTS O PEN P ROBLEMS A N E XAMPLE Consider the following tree T with vertex v : v x + x 2 + 2 x 3 + 2 x 4 + x 5 Φ T , v ( x ) =

I NTRODUCTION C OUNTING S UBTREES C OMPLEX S UBTREE R OOTS R EAL S UBTREE R OOTS O PEN P ROBLEMS A N E XAMPLE Consider the following tree T with vertex v : v x + x 2 + 2 x 3 + 2 x 4 + x 5 Φ T , v ( x ) = Φ T − v ( x ) = 4 x + 3 x 2 + 2 x 3 + x 4

I NTRODUCTION C OUNTING S UBTREES C OMPLEX S UBTREE R OOTS R EAL S UBTREE R OOTS O PEN P ROBLEMS A N E XAMPLE Consider the following tree T with vertex v : v x + x 2 + 2 x 3 + 2 x 4 + x 5 Φ T , v ( x ) = Φ T − v ( x ) = 4 x + 3 x 2 + 2 x 3 + x 4 Notice: Many of the “large” subtrees of T contain v , while many of the “small” subtrees of T do not contain v .

I NTRODUCTION C OUNTING S UBTREES C OMPLEX S UBTREE R OOTS R EAL S UBTREE R OOTS O PEN P ROBLEMS P LAN I NTRODUCTION C OUNTING S UBTREES C OMPLEX S UBTREE R OOTS R EAL S UBTREE R OOTS O PEN P ROBLEMS

I NTRODUCTION C OUNTING S UBTREES C OMPLEX S UBTREE R OOTS R EAL S UBTREE R OOTS O PEN P ROBLEMS C OUNTING S UBTREES ◮ Let s k ( T ) denote the number of subtrees of T of order k , i.e., n � s k ( T ) x k . Φ T ( x ) = k = 1

I NTRODUCTION C OUNTING S UBTREES C OMPLEX S UBTREE R OOTS R EAL S UBTREE R OOTS O PEN P ROBLEMS C OUNTING S UBTREES ◮ Let s k ( T ) denote the number of subtrees of T of order k , i.e., n � s k ( T ) x k . Φ T ( x ) = k = 1 ◮ Among all trees of order n , the star has the largest number of subtrees, while the path has the smallest number of subtrees.

I NTRODUCTION C OUNTING S UBTREES C OMPLEX S UBTREE R OOTS R EAL S UBTREE R OOTS O PEN P ROBLEMS C OUNTING S UBTREES ◮ Let s k ( T ) denote the number of subtrees of T of order k , i.e., n � s k ( T ) x k . Φ T ( x ) = k = 1 ◮ Among all trees of order n , the star has the largest number of subtrees, while the path has the smallest number of subtrees. ◮ In fact, Jamison demonstrated that for any tree T of order n ≥ 5 not isomorphic to P n or K 1 , n − 1 , � � s k ( P n ) < s k ( T ) < s k K 1 , n − 1 for all 3 ≤ k ≤ n − 1.

I NTRODUCTION C OUNTING S UBTREES C OMPLEX S UBTREE R OOTS R EAL S UBTREE R OOTS O PEN P ROBLEMS C OUNTING S UBTREES ◮ Let s k ( T ) denote the number of subtrees of T of order k , i.e., n � s k ( T ) x k . Φ T ( x ) = k = 1 ◮ Among all trees of order n , the star has the largest number of subtrees, while the path has the smallest number of subtrees. ◮ In fact, Jamison demonstrated that for any tree T of order n ≥ 5 not isomorphic to P n or K 1 , n − 1 , � � s k ( P n ) < s k ( T ) < s k K 1 , n − 1 for all 3 ≤ k ≤ n − 1. ◮ We give a short proof of this result.

I NTRODUCTION C OUNTING S UBTREES C OMPLEX S UBTREE R OOTS R EAL S UBTREE R OOTS O PEN P ROBLEMS C SIKV ´ ARI ’ S G ENERALIZED T REE S HIFT Let T be a tree, and let v 1 and v 2 be non-leaf vertices of T connected by a path whose internal vertices all have degree 2. v 1 v 2 T 1 T 2 q