. . . . . . . . . . . . . . . . General spectral graph theory: The inverse eigenvalue problem of a graph Department of Mathematics and Statistics, University of Victoria Dec 10, 2017 General spectral graph theory: IEPG 1/18 . . . . . . . . . . . . . . . . . . . . . . . . Math & Stats, University of Victoria 林晉宏 Jephian C.-H. Lin 2017 年中華民國數學年會 , Chiayi City, Taiwan
. 1 . . . . Spectral graph theory 0 1 0 . 0 1 0 1 0 0 . 2 0 1 1 2 0 2 1 2 0 2 1 General spectral graph theory: IEPG 2/18 . 1 . . . . . . . . . . . . . . . . . . Math & Stats, University of Victoria . . . . . . . . . . . . . . − 1 − 1 − 1 − 1 − 1 √ − 1 − 1 √ √ − 1 √
. 1 1 0 1 1 1 1 0 1 . Theorem (Cvetković 1971) respectively. number of positive, negative, and zero eigenvalues of A , Cvetković’s inertia bound . 1 1 . 1 3/18 General spectral graph theory: IEPG 0 0 0 1 0 1 0 0 1 1 0 0 0 . Let G be a graph and A its adjacency matrix. Then . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Math & Stats, University of Victoria . . . . The inertia of a matrix A is ( n + ( A ) , n − ( A ) , n 0 ( A )) , which are the α ( G ) ≤ min { n − n + ( A ) , n − n − ( A ) } , where α ( G ) is the independence number. n = 5 n + = 1 n − = 2
. . . . . . . . . . . . . . . . Godsil’s Lemma number of disjoint induced paths that can cover G . Theorem (Godsil 1984) Let G be a tree with adjacency matrix A. Then General spectral graph theory: IEPG 4/18 . . . . . . . . . . . . . . . . . . . . . . . . Math & Stats, University of Victoria Let G be a graph. The path cover number P ( G ) is the minimum m λ ( A ) ≤ P ( G ) for any eigenvalue λ of A. m 0 ( A ) = 4
. . . . . . . . . . . . . . . General spectral graph theory real symmetric matrices M with and so on. 0 0 General spectral graph theory: IEPG 5/18 . . . . . . . . . . . . . . Math & Stats, University of Victoria . . . . . . . . . . . Given a grpah G on n vertices, consider the family S ( G ) of n × n if i ̸ = j and { i , j } is not an edge , M i , j = 0 M i , j ̸ = 0 if i ̸ = j and { i , j } is an edge , M i , j ∈ R if i = j . Thus, S ( G ) includes the adjacency matrix, the Laplacian matrix, ? ∗ ∗ ? ∗ ∗ ?
. . . . . . . . . . . . . . . The general version of Cvetković’s inertia bound Theorem inertia bound is not tight. (So far, all known constructions are related to Paley 17.) Diego! General spectral graph theory: IEPG 6/18 . . . . . . . . . . . . . . . . . . . . . Math & Stats, University of Victoria . . . . Let G be a graph and A ∈ S ( G ) with zero diagonal entries. Then α ( G ) ≤ min { n − n + ( A ) , n − n − ( A ) } , where α ( G ) is the independence number. ▶ Sinkovic (2017) proved Paley 17 is an example where the ▶ He is going to talk about it at the Joint Meeting 2018 in San
. . . . . . . . . . . . . . . The general version of Cvetković’s inertia bound Theorem inertia bound is not tight. (So far, all known constructions are related to Paley 17.) Diego! General spectral graph theory: IEPG 6/18 . . . . . . . . . . . . . . . . . . . . . Math & Stats, University of Victoria . . . . Let G be a graph and A ∈ S ( G ) with zero diagonal entries. Then α ( G ) ≤ min { n − n + ( A ) , n − n − ( A ) } , where α ( G ) is the independence number. ▶ Sinkovic (2017) proved Paley 17 is an example where the ▶ He is going to talk about it at the Joint Meeting 2018 in San
. . . . . . . . . . . . . . . . . The general version of Godsil’s lemma Theorem (Johnson and Leal Duarte 1999) General spectral graph theory: IEPG 7/18 . . . . . . . . . . . . . . . . . . . . . . . Math & Stats, University of Victoria Let G be a tree and A ∈ S ( G ) . Then m λ ( A ) ≤ P ( G ) for any eigenvalue λ of A. ▶ Indeed, for any tree, there is a matrix A with an eigenvalue λ such that m λ ( A ) = P ( G ) .
. . . . . . . . . . . . . . . . . The general version of Godsil’s lemma Theorem (Johnson and Leal Duarte 1999) General spectral graph theory: IEPG 7/18 . . . . . . . . . . . . . . . . . . . . . . . Math & Stats, University of Victoria Let G be a tree and A ∈ S ( G ) . Then m λ ( A ) ≤ P ( G ) for any eigenvalue λ of A. ▶ Indeed, for any tree, there is a matrix A with an eigenvalue λ such that m λ ( A ) = P ( G ) .
. . . . . . . . . . . . . . . . Domination number cardinality of a set X such that a set X such that General spectral graph theory: IEPG 8/18 . . . . . . . . . . . . . . . . . . . . Math & Stats, University of Victoria . . . . Let G be a graph. The domination number γ ( G ) is the minimum ∪ N G [ x ] = V ( G ) . x ∈ X The total domination number γ t ( G ) is the minimum cardinality of ∪ N G ( x ) = V ( G ) . x ∈ X γ ( P 3 ) = 1 γ t ( P 3 ) = 2
. . . . . . . . . . . . . . . Greedy algorithm making the locally optimal choice at each stage with the hope of fjnding a global optimum. it might lead you to a dead end. and not yet dominate the whole graph, pick a vertex v such that General spectral graph theory: IEPG 9/18 . . . . . . . . . . . . . . Math & Stats, University of Victoria . . . . . . . . . . . ▶ Greedy algorithm follows the problem solving heuristic of ▶ For solving a maze, you may keep going straight at fork. But ▶ For a coloring problem, you may keep using the smallest free number to color the next vertex, showing χ ( G ) ≤ ∆( G ) + 1. ▶ Greedy algorithm for domination number: When X are chosen ∪ N G [ v ] \ N G [ x ] ̸ = ∅ . x ∈ X
. . . . . . . . . . . . . . . Greedy algorithm making the locally optimal choice at each stage with the hope of fjnding a global optimum. it might lead you to a dead end. and not yet dominate the whole graph, pick a vertex v such that General spectral graph theory: IEPG 9/18 . . . . . . . . . . . . . . Math & Stats, University of Victoria . . . . . . . . . . . ▶ Greedy algorithm follows the problem solving heuristic of ▶ For solving a maze, you may keep going straight at fork. But ▶ For a coloring problem, you may keep using the smallest free number to color the next vertex, showing χ ( G ) ≤ ∆( G ) + 1. ▶ Greedy algorithm for domination number: When X are chosen ∪ N G [ v ] \ N G [ x ] ̸ = ∅ . x ∈ X
. . . . . . . . . . . . . . . Greedy algorithm making the locally optimal choice at each stage with the hope of fjnding a global optimum. it might lead you to a dead end. and not yet dominate the whole graph, pick a vertex v such that General spectral graph theory: IEPG 9/18 . . . . . . . . . . . . . . Math & Stats, University of Victoria . . . . . . . . . . . ▶ Greedy algorithm follows the problem solving heuristic of ▶ For solving a maze, you may keep going straight at fork. But ▶ For a coloring problem, you may keep using the smallest free number to color the next vertex, showing χ ( G ) ≤ ∆( G ) + 1. ▶ Greedy algorithm for domination number: When X are chosen ∪ N G [ v ] \ N G [ x ] ̸ = ∅ . x ∈ X
. . . . . . . . . . . . . . . Greedy algorithm making the locally optimal choice at each stage with the hope of fjnding a global optimum. it might lead you to a dead end. and not yet dominate the whole graph, pick a vertex v such that General spectral graph theory: IEPG 9/18 . . . . . . . . . . . . . . Math & Stats, University of Victoria . . . . . . . . . . . ▶ Greedy algorithm follows the problem solving heuristic of ▶ For solving a maze, you may keep going straight at fork. But ▶ For a coloring problem, you may keep using the smallest free number to color the next vertex, showing χ ( G ) ≤ ∆( G ) + 1. ▶ Greedy algorithm for domination number: When X are chosen ∪ N G [ v ] \ N G [ x ] ̸ = ∅ . x ∈ X
. . . . . . . . . . . . . . . . . Grundy domination number General spectral graph theory: IEPG 10/18 . . . . . . . . . . . . . . . . . . Math & Stats, University of Victoria . . . . . The Grundy domination number γ gr ( G ) is the length of the longest sequence ( v 1 , v 2 , . . . , v k ) such that i − 1 ∪ N G [ v i ] \ N G [ v j ] ̸ = ∅ . j = 1
. . . . . . . . . . . . . . . . . Grundy domination number General spectral graph theory: IEPG 10/18 . . . . . . . . . . . . . . . . . . Math & Stats, University of Victoria . . . . . The Grundy domination number γ gr ( G ) is the length of the longest sequence ( v 1 , v 2 , . . . , v k ) such that i − 1 ∪ N G [ v i ] \ N G [ v j ] ̸ = ∅ . j = 1
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