Average Connectivity and Average Edge-connectivity in Graphs Suil O - PowerPoint PPT Presentation

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs Average Connectivity and Average Edge-connectivity in Graphs Suil O joint work with Jaehoon Kim University of Illinois at Urbana-Champaign 24th Cumberland Conference on Combinatorics, Graph Theory, and Computing (May 12th 2011) Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs Table of Contents Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs Connectivity and Edge-connectivity ◮ The connectivity of G , written κ ( G ), is the minimum size of a vertex set S such that G − S is disconnected. Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs Connectivity and Edge-connectivity ◮ The connectivity of G , written κ ( G ), is the minimum size of a vertex set S such that G − S is disconnected. ◮ The edge-connectivity of G , written κ ′ ( G ), is the minimum size of an edge set F such that G − F is disconnected. Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs Connectivity and Edge-connectivity ◮ The connectivity of G , written κ ( G ), is the minimum size of a vertex set S such that G − S is disconnected. ◮ The edge-connectivity of G , written κ ′ ( G ), is the minimum size of an edge set F such that G − F is disconnected. The connectivity and the edge-connectivity of a graph measure the difficulty of breaking the graph apart. Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs Connectivity and Edge-connectivity ◮ The connectivity of G , written κ ( G ), is the minimum size of a vertex set S such that G − S is disconnected. ◮ The edge-connectivity of G , written κ ′ ( G ), is the minimum size of an edge set F such that G − F is disconnected. The connectivity and the edge-connectivity of a graph measure the difficulty of breaking the graph apart. However, since these values are based on a worst-case situation, it does not reflect the “global (edge) connectedness” of the graph. Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs Connectivity and Edge-connectivity ◮ The connectivity of G , written κ ( G ), is the minimum size of a vertex set S such that G − S is disconnected. ◮ The edge-connectivity of G , written κ ′ ( G ), is the minimum size of an edge set F such that G − F is disconnected. The connectivity and the edge-connectivity of a graph measure the difficulty of breaking the graph apart. However, since these values are based on a worst-case situation, it does not reflect the “global (edge) connectedness” of the graph. Figure: TwoGraphs G 1 and G 2 with connectivity 1 and edge-connectivity 1 Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs Average Connectivity and Average Edge-connectivity The average connectivity of a graph G with n vertices, written P u , v ∈ V ( G ) κ ( u , v ) κ ( G ), is , where κ ( u , v ) is the minimum number of ( n 2 ) vertices whose deletion makes v unreachable from u . Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs Average Connectivity and Average Edge-connectivity The average connectivity of a graph G with n vertices, written P u , v ∈ V ( G ) κ ( u , v ) κ ( G ), is , where κ ( u , v ) is the minimum number of ( n 2 ) vertices whose deletion makes v unreachable from u . The average edge-connectivity of a graph G with n vertices, P u , v ∈ V ( G ) κ ′ ( u , v ) , where κ ′ ( u , v ) is the minimum written κ ′ ( G ), is ( n 2 ) number of edges whose deletion makes v unreachable from u . Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs Average Connectivity and Average Edge-connectivity The average connectivity of a graph G with n vertices, written P u , v ∈ V ( G ) κ ( u , v ) κ ( G ), is , where κ ( u , v ) is the minimum number of ( n 2 ) vertices whose deletion makes v unreachable from u . The average edge-connectivity of a graph G with n vertices, P u , v ∈ V ( G ) κ ′ ( u , v ) , where κ ′ ( u , v ) is the minimum written κ ′ ( G ), is ( n 2 ) number of edges whose deletion makes v unreachable from u . Figure: κ ( G 1 ) = κ ′ ( G 1 ) = 27 7 and κ ( G 2 ) = κ ′ ( G 2 ) = 12 7 Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs Average Connectivity and Matching Number In 2002, Beineke, Oellermann and Pippert introduced the average connectivity and found several properties of it. Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs Average Connectivity and Matching Number In 2002, Beineke, Oellermann and Pippert introduced the average connectivity and found several properties of it. Theorem (Dankelmann and Oellermann 2003) 2 d If G has average degree d and n vertices, then n − 1 ≤ κ ( G ) ≤ d . Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs Average Connectivity and Matching Number In 2002, Beineke, Oellermann and Pippert introduced the average connectivity and found several properties of it. Theorem (Dankelmann and Oellermann 2003) 2 d If G has average degree d and n vertices, then n − 1 ≤ κ ( G ) ≤ d . We prove a bound on the average connectivity in terms of matching number. Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs Average Connectivity and Matching Number In 2002, Beineke, Oellermann and Pippert introduced the average connectivity and found several properties of it. Theorem (Dankelmann and Oellermann 2003) 2 d If G has average degree d and n vertices, then n − 1 ≤ κ ( G ) ≤ d . We prove a bound on the average connectivity in terms of matching number. Theorem (Kim and O 2011++) For a connected graph G , κ ( G ) ≤ 2 α ′ ( G ), and this is sharp. Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs Average Connectivity and Matching Number In 2002, Beineke, Oellermann and Pippert introduced the average connectivity and found several properties of it. Theorem (Dankelmann and Oellermann 2003) 2 d If G has average degree d and n vertices, then n − 1 ≤ κ ( G ) ≤ d . We prove a bound on the average connectivity in terms of matching number. Theorem (Kim and O 2011++) For a connected graph G , κ ( G ) ≤ 2 α ′ ( G ), and this is sharp. Furthermore, if G is connected and bipartite, � � 9 3 n − 4 α ′ ( G ), and this is sharp. then κ ( G ) ≤ 8 − 8 n 2 − 8 n Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs Proof (Average Connectivity and Matching Number) Theorem (Kim and O 2011++) For a connected graph G , k ( G ) ≤ 2 α ′ ( G ). Thisissharponlyforcomplete graphswithan odd number of vertices. Suil O Average Connectivity and Average Edge-connectivity in Graphs

Average Connectivity and Average Edge-connectivity Average Connectivity and Matching Average Edge-connectivity in Regular Graphs Proof (Average Connectivity and Matching Number) Theorem (Kim and O 2011++) For a connected graph G , k ( G ) ≤ 2 α ′ ( G ). Thisissharponlyforcomplete graphswithan odd number of vertices. Proof: Suil O Average Connectivity and Average Edge-connectivity in Graphs