OPERATIONS ON GRAPHS INCREASING SOME GRAPH PARAMETERS Alexander - PowerPoint PPT Presentation

OPERATIONS ON GRAPHS INCREASING SOME GRAPH PARAMETERS Alexander Kelmans University of Puerto Rico Rutgers University May 25, 2014

1. Let G m n be the set of graphs with n vertices and m edges. Let Q be an operation on a graph such that G ∈ G m ⇒ Q ( G ) ∈ G m n . n

1. Let G m n be the set of graphs with n vertices and m edges. Let Q be an operation on a graph such that G ∈ G m ⇒ Q ( G ) ∈ G m n . n 2. Let ( G m n , � ) be a quasi-poset. An operation Q is called � -increasing ( � -decreasing ) if (resp., Q ( G ) � G ) for every G ∈ G m Q ( G ) � G n .

1. A graph G is called vertex comparable (A.K. 1970) if N ( x , G ) \ x ⊆ N ( y , G ) \ y or N ( y , G ) \ y ⊆ N ( x , G ) \ x for every x , y ∈ V ( G ).

1. A graph G is called vertex comparable (A.K. 1970) if N ( x , G ) \ x ⊆ N ( y , G ) \ y or N ( y , G ) \ y ⊆ N ( x , G ) \ x for every x , y ∈ V ( G ). 2. A graph G is called threshold (V. Chv´ atal, P. Hammer 1973) if G has no induced � , N or II.

1. A graph G is called vertex comparable (A.K. 1970) if N ( x , G ) \ x ⊆ N ( y , G ) \ y or N ( y , G ) \ y ⊆ N ( x , G ) \ x for every x , y ∈ V ( G ). 2. A graph G is called threshold (V. Chv´ atal, P. Hammer 1973) if G has no induced � , N or II. 3. Claim. G is vertex comparable if and only if G is threshold.

1. Let k , r , s be integers, k ≥ 0, and 0 ≤ r < s . Let F ( k , r , s ) be the graph obtained from the complete graph K s as follows: • fix in K s a set A of r vertices and a ∈ A , • add to K s a new vertex c and the set { cx : x ∈ A } of new edges to obtain graph C ( r , s ), and • add to C ( r , s ) the set B of k new vertices and the set { az : z ∈ B } of new edge to obtain graph F ( k , r , s ).

1. Let k , r , s be integers, k ≥ 0, and 0 ≤ r < s . Let F ( k , r , s ) be the graph obtained from the complete graph K s as follows: • fix in K s a set A of r vertices and a ∈ A , • add to K s a new vertex c and the set { cx : x ∈ A } of new edges to obtain graph C ( r , s ), and • add to C ( r , s ) the set B of k new vertices and the set { az : z ∈ B } of new edge to obtain graph F ( k , r , s ). 2. Let C m n be the set of connected graphs with n vertices and m edges. For every pair ( n , m ) of integers such that C m Claim. n � = ∅ there exists a unique triple ( k , r , s ) of integers such that k ≥ 0 , 0 ≤ r < s, and F ( k , r , s ) ∈ C m n . We call F ( k , r , s ) = F m n the extreme graph in C m n .

1. Theorem (A.K. 1970). Let n and m be natural numbers and n ≥ 3 . ( a 1) If n − 1 ≤ m ≤ 2 n − 4 , then F m n is the only threshold graph with n vertices and m edges, i.e. F m n = { F m n } . ( a 2) If m = 2 n − 3 , then F m n is not the only threshold graph with n vertices and m edges.

1. Theorem (A.K. 1970). Let n and m be natural numbers and n ≥ 3 . ( a 1) If n − 1 ≤ m ≤ 2 n − 4 , then F m n is the only threshold graph with n vertices and m edges, i.e. F m n = { F m n } . ( a 2) If m = 2 n − 3 , then F m n is not the only threshold graph with n vertices and m edges. 2. Theorem (A.K. 1970). Let G be a connected graph. Then ( a 1) there exists a connected threshold graph F obtained from G by a series of ♦ -operations, and so ( a 2) if the ♦ -operation is � -decreasing, then there exists a connected threshold graph F such that G � F.

1. Theorem (A.K. 1970). Let the ♦ -operation be � -decreasing, F a graph with r edges and at most n vertices, and rP 1 and S r are a matching and a star with r edges. Then K n − E ( rP 1 ) � K n − E ( S 2 +( r − 2) P 1 ) � K n − E ( F ) � K n − E ( S r ) , where r ≥ 2 , r ≤ n / 2 for the second � , and r ≤ n − 1 for the last � .

1. Theorem (A.K. 1970). Let the ♦ -operation be � -decreasing, F a graph with r edges and at most n vertices, and rP 1 and S r are a matching and a star with r edges. Then K n − E ( rP 1 ) � K n − E ( S 2 +( r − 2) P 1 ) � K n − E ( F ) � K n − E ( S r ) , where r ≥ 2 , r ≤ n / 2 for the second � , and r ≤ n − 1 for the last � . 2. Theorem (A.K. 1970). Let the ♦ -operation be � -decreasing, n , and ¨ G ∈ C m G be obtained from G by adding m − n + 1 isolated vertices. Then for every spanning tree T of G there exists a tree D with m edges such that T is a subgraph of D and D � ¨ G.

Theorem (A.K. 1970) Let G ∈ C m n , P n an n-vertex path, C n an n-vertex cycle, and G �∈ F m n . Suppose that the ♦ -operation is � -decreasing. ( a 1) If m = n − 1 ≥ 3 and G � = P n , then P n ≻ G ≻ F n − 1 . n ( a 2) If m = n ≥ 3 and G � = C n , then C n ≻ G ≻ F n n . G ≻ F n +1 ( a 3) If n ≥ 4 and m = n + 1 , then . n ( a 4) If n ≥ 5 and n + 2 ≤ m ≤ 2 n − 4 , then G ≻ F m n .

1. Suppose that every edge of a graph G has probability p to exist and the edge events are independent.

1. Suppose that every edge of a graph G has probability p to exist and the edge events are independent. 2. Let R ( p , G ) be the probability that the random graph ( G , p ) is connected. We call R ( p , G ) the the reliability of G .

1. Suppose that every edge of a graph G has probability p to exist and the edge events are independent. 2. Let R ( p , G ) be the probability that the random graph ( G , p ) is connected. We call R ( p , G ) the the reliability of G . 3. Then { a s ( G ) p s q m − s : s ∈ { n − 1 , . . . , m } } , � R ( p , G ) = where n and m are the numbers of vertices and edges of G , q = 1 − p , and a s ( G ) is the number of connected spanning subgraphs of G with s edges, and so a n − 1 = t ( G ) is the number of spanning trees of G .

1. Problem Find a most reliable graph M ( p ) with n vertices and m edges, i.e. such that R ( p , M ( p )) = max { R ( p , G ) : G ∈ G m n } .

1. Problem Find a most reliable graph M ( p ) with n vertices and m edges, i.e. such that R ( p , M ( p )) = max { R ( p , G ) : G ∈ G m n } . 2. Problem Find a least reliable connected graph L ( p ) with n vertices and m edges, i.e. such that R ( p , L ( p )) = min { R ( p , G ) : G ∈ C m n } .

1. Problem Find a most reliable graph M ( p ) with n vertices and m edges, i.e. such that R ( p , M ( p )) = max { R ( p , G ) : G ∈ G m n } . 2. Problem Find a least reliable connected graph L ( p ) with n vertices and m edges, i.e. such that R ( p , L ( p )) = min { R ( p , G ) : G ∈ C m n } . Find a graph A m n ∈ G m 3. Problem n with the maximum number a s ( G ) of connected spanning subgraps with s edges: a s ( F m n ) = max { a s ( G ) : G ∈ G m n } .

1. Problem Find a most reliable graph M ( p ) with n vertices and m edges, i.e. such that R ( p , M ( p )) = max { R ( p , G ) : G ∈ G m n } . 2. Problem Find a least reliable connected graph L ( p ) with n vertices and m edges, i.e. such that R ( p , L ( p )) = min { R ( p , G ) : G ∈ C m n } . Find a graph A m n ∈ G m 3. Problem n with the maximum number a s ( G ) of connected spanning subgraps with s edges: a s ( F m n ) = max { a s ( G ) : G ∈ G m n } . Find a graph B m ∈ G m 4. Problem n with the maximum number n of spanning trees, i.e. such that t ( B m n ) = max { t ( G ) : G ∈ G m n } .

1. Poset ( G m n , � r ): G � r F if R ( p , G ) ≥ R ( p , F ) for every p ∈ [0 , 1] .

1. Poset ( G m n , � r ): G � r F if R ( p , G ) ≥ R ( p , F ) for every p ∈ [0 , 1] . 2. Poset ( G m n , � a ): G � a F if a s ( G ) ≥ a s ( F ) for s ∈ { n − 1 , . . . , m } .

1. Poset ( G m n , � r ): G � r F if R ( p , G ) ≥ R ( p , F ) for every p ∈ [0 , 1] . 2. Poset ( G m n , � a ): G � a F if a s ( G ) ≥ a s ( F ) for s ∈ { n − 1 , . . . , m } . 3. Poset ( G m n , � t ): G � t F if t ( G ) ≥ t ( F ) .

1. Poset ( G m n , � r ): G � r F if R ( p , G ) ≥ R ( p , F ) for every p ∈ [0 , 1] . 2. Poset ( G m n , � a ): G � a F if a s ( G ) ≥ a s ( F ) for s ∈ { n − 1 , . . . , m } . 3. Poset ( G m n , � t ): G � t F if t ( G ) ≥ t ( F ) . � t . � a � r 4. Obviously, ⇒ ⇒

1. Poset ( G m n , � r ): G � r F if R ( p , G ) ≥ R ( p , F ) for every p ∈ [0 , 1] . 2. Poset ( G m n , � a ): G � a F if a s ( G ) ≥ a s ( F ) for s ∈ { n − 1 , . . . , m } . 3. Poset ( G m n , � t ): G � t F if t ( G ) ≥ t ( F ) . � t . � a � r 4. Obviously, ⇒ ⇒ ∈ {� a , � r , � t } . 5. Theorem (A.K. 1966) Let � Let G , G ′ ∈ G m n and G ′ = H xy ( G ). Then G � G ′ .

1. Let L ( λ, G ) be the characteristic polynomial of the Laplacian matrix of G .

1. Let L ( λ, G ) be the characteristic polynomial of the Laplacian matrix of G . 2. Poset ( G m n , � L ): G � L F if L ( λ, G ) ≥ L ( λ, F ) for λ ≥ n .

1. Let L ( λ, G ) be the characteristic polynomial of the Laplacian matrix of G . 2. Poset ( G m n , � L ): G � L F if L ( λ, G ) ≥ L ( λ, F ) for λ ≥ n . n , � τ ) ): 3. Poset ( G m G � τ F if t ( K n + r − E ( G )) ≥ t ( K n + r − E ( F )) for integer r ≥ 0 .

1. Let L ( λ, G ) be the characteristic polynomial of the Laplacian matrix of G . 2. Poset ( G m n , � L ): G � L F if L ( λ, G ) ≥ L ( λ, F ) for λ ≥ n . n , � τ ) ): 3. Poset ( G m G � τ F if t ( K n + r − E ( G )) ≥ t ( K n + r − E ( F )) for integer r ≥ 0 . � τ . 4. Claim (A.K. 1965). � L ⇒