❆❞❥❛❝❡♥❝② ♠❛tr✐❝❡s ♦❢ ❝♦♠♣❧❡① ✉♥✐t ❣❛✐♥ ❣r❛♣❤s M. Rajesh Kannan Department of Mathematics, Indian Institute of Technology Kharagpur, email: rajeshkannan1.m@gmail.com, rajeshkannan@maths.iitkgp.ac.in August 21, 2020 1/25
Outline Adjacency matrices of graphs Spectral properties Perron-Frobenius theorem Adjacency matrices of complex unit gain graphs Characterization of bipartite graphs and trees 2/25
Adjacency matrix Definition (Adjacency matrix) The adjacency matrix of a graph G with n vertices, V ( G ) = { v 1 , . . . , v n } is the n × n matrix, denoted by A ( G ) = ( a ij ) , is defined by � 1 if v i ∼ v j , a ij = 0 otherwise. 3/25
Example Example Consider the graph G 4 3 2 1 4/25
Example Example Consider the graph G 4 3 2 1 The adjacency matrix of G is 0 1 1 1 1 0 1 0 A ( G ) = 1 1 0 0 1 0 0 0 4/25
Properties Let G be a connected graph with vertices { v 1 , v 2 , . . . , v n } and let A be the adjacency matrix of G . 5/25
Properties Let G be a connected graph with vertices { v 1 , v 2 , . . . , v n } and let A be the adjacency matrix of G . Then, A is symmetric. 1 5/25
Properties Let G be a connected graph with vertices { v 1 , v 2 , . . . , v n } and let A be the adjacency matrix of G . Then, A is symmetric. 1 Sum of the 2 × 2 principal minors of A equals to −| E ( G ) | . 2 5/25
Properties Let G be a connected graph with vertices { v 1 , v 2 , . . . , v n } and let A be the adjacency matrix of G . Then, A is symmetric. 1 Sum of the 2 × 2 principal minors of A equals to −| E ( G ) | . 2 ( i , j ) th entry of the matrix A k equals the number of walks of length 3 k from the vertex i to the vertex j . 5/25
Spectrum of adjacency matrix Let G be a graph with n vertices and with eigenvalues of its adjacency matrices, λ 1 ≥ λ 2 ≥ · · · ≥ λ n . We denote by ∆( G ) and δ ( G ) , the maximum and the minimum of the vertex degrees of G , respectively. 6/25
Spectrum of adjacency matrix Let G be a graph with n vertices and with eigenvalues of its adjacency matrices, λ 1 ≥ λ 2 ≥ · · · ≥ λ n . We denote by ∆( G ) and δ ( G ) , the maximum and the minimum of the vertex degrees of G , respectively. Properties of spectrum δ ( G ) ≤ λ 1 ≤ ∆( G ) . 6/25
Spectrum of adjacency matrix Let G be a graph with n vertices and with eigenvalues of its adjacency matrices, λ 1 ≥ λ 2 ≥ · · · ≥ λ n . We denote by ∆( G ) and δ ( G ) , the maximum and the minimum of the vertex degrees of G , respectively. Properties of spectrum δ ( G ) ≤ λ 1 ≤ ∆( G ) . χ ( G ) ≤ 1 + λ 1 , where χ ( G ) is the chromatic number of G . 6/25
Spectrum of adjacency matrix Let G be a graph with n vertices and with eigenvalues of its adjacency matrices, λ 1 ≥ λ 2 ≥ · · · ≥ λ n . We denote by ∆( G ) and δ ( G ) , the maximum and the minimum of the vertex degrees of G , respectively. Properties of spectrum δ ( G ) ≤ λ 1 ≤ ∆( G ) . χ ( G ) ≤ 1 + λ 1 , where χ ( G ) is the chromatic number of G . χ ( G ) ≥ 1 − λ 1 λ n . 6/25
Spectrum of adjacency matrix Let G be a graph with n vertices and with eigenvalues of its adjacency matrices, λ 1 ≥ λ 2 ≥ · · · ≥ λ n . We denote by ∆( G ) and δ ( G ) , the maximum and the minimum of the vertex degrees of G , respectively. Properties of spectrum δ ( G ) ≤ λ 1 ≤ ∆( G ) . χ ( G ) ≤ 1 + λ 1 , where χ ( G ) is the chromatic number of G . χ ( G ) ≥ 1 − λ 1 λ n . G is bipartite if and only if the eigenvalues of A are symmetric with respect to origin. 6/25
Spectrum of adjacency matrix Let G be a graph with n vertices and with eigenvalues of its adjacency matrices, λ 1 ≥ λ 2 ≥ · · · ≥ λ n . We denote by ∆( G ) and δ ( G ) , the maximum and the minimum of the vertex degrees of G , respectively. Properties of spectrum δ ( G ) ≤ λ 1 ≤ ∆( G ) . χ ( G ) ≤ 1 + λ 1 , where χ ( G ) is the chromatic number of G . χ ( G ) ≥ 1 − λ 1 λ n . G is bipartite if and only if the eigenvalues of A are symmetric with respect to origin. That is, λ is an eigenvalue of A ( G ) if and only if − λ is an eigenvalue of A ( G ) . 6/25
Irreducible matrices An n × n matrix, n ≥ 2, is reducible its rows and columns can be simultaneously permuted to � B � C 0 D where B and D are square (not necessarily of the same order). 7/25
Irreducible matrices An n × n matrix, n ≥ 2, is reducible its rows and columns can be simultaneously permuted to � B � C 0 D where B and D are square (not necessarily of the same order). Otherwise, it is irreducible . For n = 1, 0 is reducible, a � = 0 is irreducible. 7/25
Irreducible matrices An n × n matrix, n ≥ 2, is reducible its rows and columns can be simultaneously permuted to � B � C 0 D where B and D are square (not necessarily of the same order). Otherwise, it is irreducible . For n = 1, 0 is reducible, a � = 0 is irreducible. The directed graph G ( A ) , associated with an n × n matrix has n vertices 1 , . . . , n and an arc from i to j if and only if a ij � = 0 . 7/25
Irreducible matrices An n × n matrix, n ≥ 2, is reducible its rows and columns can be simultaneously permuted to � B � C 0 D where B and D are square (not necessarily of the same order). Otherwise, it is irreducible . For n = 1, 0 is reducible, a � = 0 is irreducible. The directed graph G ( A ) , associated with an n × n matrix has n vertices 1 , . . . , n and an arc from i to j if and only if a ij � = 0 . Working definition: A is irreducible if and only if G ( A ) is strongly connected. 7/25
Example 1 0 1 0 0 0 0 1 0 1 0 0 1 0 0 1 8/25
Example 1 2 4 3 1 0 1 0 0 0 0 1 0 1 0 0 1 0 0 1 8/25
Example 1 0 1 0 0 1 1 1 1 0 1 0 1 1 0 1 9/25
Example 1 2 4 3 1 0 1 0 0 1 1 1 1 0 1 0 1 1 0 1 9/25
Perron-Frobenius Theorem Theorem If A is nonnegative and irreducible, then a) ρ ( A ) > 0 , where ρ ( A ) is the maximum of absolute value of all the eigenvalues of A, b) ρ ( A ) is an eigenvalue of A, c) There is a positive vector such that Ax = ρ ( A ) x. 10/25
Perron-Frobenius Theorem Theorem If A is nonnegative and irreducible, then a) ρ ( A ) > 0 , where ρ ( A ) is the maximum of absolute value of all the eigenvalues of A, b) ρ ( A ) is an eigenvalue of A, c) There is a positive vector such that Ax = ρ ( A ) x. Theorem Let A , B ∈ C n × n and suppose that A is nonnegative. If A ≥ | B | , then ρ ( A ) ≥ ρ ( | B | ) ≥ ρ ( B ) . 10/25
Perron-Frobenius Theorem Theorem If A is nonnegative and irreducible, then a) ρ ( A ) > 0 , where ρ ( A ) is the maximum of absolute value of all the eigenvalues of A, b) ρ ( A ) is an eigenvalue of A, c) There is a positive vector such that Ax = ρ ( A ) x. Theorem Let A , B ∈ C n × n and suppose that A is nonnegative. If A ≥ | B | , then ρ ( A ) ≥ ρ ( | B | ) ≥ ρ ( B ) . Theorem Let A , B ∈ C n × n . Suppose A is nonnegative and irreducible, and A ≥ | B | . Let λ = e i θ ρ ( B ) be a maximum-modulus eigenvalue of B. 10/25
Perron-Frobenius Theorem Theorem If A is nonnegative and irreducible, then a) ρ ( A ) > 0 , where ρ ( A ) is the maximum of absolute value of all the eigenvalues of A, b) ρ ( A ) is an eigenvalue of A, c) There is a positive vector such that Ax = ρ ( A ) x. Theorem Let A , B ∈ C n × n and suppose that A is nonnegative. If A ≥ | B | , then ρ ( A ) ≥ ρ ( | B | ) ≥ ρ ( B ) . Theorem Let A , B ∈ C n × n . Suppose A is nonnegative and irreducible, and A ≥ | B | . Let λ = e i θ ρ ( B ) be a maximum-modulus eigenvalue of B. If ρ ( A ) = ρ ( B ) , then there is a diagonal unitary matrix D ∈ C n × n such that B = e i θ DAD − 1 . 10/25
Gain graphs Let G be a group and, let G be a simple graph with vertex set V ( G ) = { 1 , 2 , . . . , n } and edge set E ( G ) = { e 1 , . . . , e m } . 11/25
Gain graphs Let G be a group and, let G be a simple graph with vertex set V ( G ) = { 1 , 2 , . . . , n } and edge set E ( G ) = { e 1 , . . . , e m } . Define e jk as a directed edge from the vertex j to the vertex k , if there is an edge between them. 11/25
Gain graphs Let G be a group and, let G be a simple graph with vertex set V ( G ) = { 1 , 2 , . . . , n } and edge set E ( G ) = { e 1 , . . . , e m } . Define e jk as a directed edge from the vertex j to the vertex k , if there is an edge between them. The directed edge set − − − → E ( G ) consists of the directed edges e jk , e kj ∈ − − − → E ( G ) , for each adjacent vertices j and k of G . 11/25
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