Adjacency Matrices

This module defines the adjacency matrix of a graph, and provides theorems connecting graph properties to computational properties of the matrix.

Main definitions

• Matrix.IsAdjMatrix: A : Matrix V V α is qualified as an "adjacency matrix" if (1) every entry of A is 0 or 1, (2) A is symmetric, (3) every diagonal entry of A is 0.

• Matrix.IsAdjMatrix.to_graph: for A : Matrix V V α and h : A.IsAdjMatrix, h.to_graph is the simple graph induced by A.

• Matrix.compl: for A : Matrix V V α, A.compl is supposed to be the adjacency matrix of the complement graph of the graph induced by A.

• SimpleGraph.adjMatrix: the adjacency matrix of a SimpleGraph.

• SimpleGraph.adjMatrix_pow_apply_eq_card_walk: each entry of the nth power of a graph's adjacency matrix counts the number of length-n walks between the corresponding pair of vertices.

structure Matrix.IsAdjMatrix {V : Type u_1} {α : Type u_2} [Zero α] [One α] (A : Matrix V V α) : Prop

A : Matrix V V α is qualified as an "adjacency matrix" if (1) every entry of A is 0 or 1, (2) A is symmetric, (3) every diagonal entry of A is 0.

• zero_or_one (i j : V) : A i j = 0 ∨ A i j = 1
• symm : A.IsSymm
• apply_diag (i : V) : A i i = 0

@[simp]

theorem Matrix.IsAdjMatrix.apply_diag_ne {V : Type u_1} {α : Type u_2} {A : Matrix V V α} [MulZeroOneClass α] [Nontrivial α] (h : A.IsAdjMatrix) (i : V) : ¬A i i = 1

@[simp]

theorem Matrix.IsAdjMatrix.apply_ne_one_iff {V : Type u_1} {α : Type u_2} {A : Matrix V V α} [MulZeroOneClass α] [Nontrivial α] (h : A.IsAdjMatrix) (i j : V) : ¬A i j = 1 ↔ A i j = 0

@[simp]

theorem Matrix.IsAdjMatrix.apply_ne_zero_iff {V : Type u_1} {α : Type u_2} {A : Matrix V V α} [MulZeroOneClass α] [Nontrivial α] (h : A.IsAdjMatrix) (i j : V) : ¬A i j = 0 ↔ A i j = 1

def Matrix.IsAdjMatrix.toGraph {V : Type u_1} {α : Type u_2} {A : Matrix V V α} [MulZeroOneClass α] [Nontrivial α] (h : A.IsAdjMatrix) : SimpleGraph V

For A : Matrix V V α and h : IsAdjMatrix A, h.toGraph is the simple graph whose adjacency matrix is A.

Equations

• h.toGraph = { Adj := fun (i j : V) => A i j = 1, symm := ⋯, loopless := ⋯ }

@[simp]

theorem Matrix.IsAdjMatrix.toGraph_adj {V : Type u_1} {α : Type u_2} {A : Matrix V V α} [MulZeroOneClass α] [Nontrivial α] (h : A.IsAdjMatrix) (i j : V) : h.toGraph.Adj i j = (A i j = 1)

instance Matrix.IsAdjMatrix.instDecidableRelAdjToGraphOfDecidableEq {V : Type u_1} {α : Type u_2} {A : Matrix V V α} [MulZeroOneClass α] [Nontrivial α] [DecidableEq α] (h : A.IsAdjMatrix) : DecidableRel h.toGraph.Adj

Equations

• h.instDecidableRelAdjToGraphOfDecidableEq = id inferInstance

def Matrix.compl {V : Type u_1} {α : Type u_2} [Zero α] [One α] [DecidableEq α] [DecidableEq V] (A : Matrix V V α) : Matrix V V α

For A : Matrix V V α, A.compl is supposed to be the adjacency matrix of the complement graph of the graph induced by A.adjMatrix.

Equations

• A.compl i j = if i = j then 0 else if A i j = 0 then 1 else 0

@[simp]

theorem Matrix.compl_apply_diag {V : Type u_1} {α : Type u_2} [DecidableEq α] [DecidableEq V] (A : Matrix V V α) [Zero α] [One α] (i : V) : A.compl i i = 0

@[simp]

theorem Matrix.compl_apply {V : Type u_1} {α : Type u_2} [DecidableEq α] [DecidableEq V] (A : Matrix V V α) [Zero α] [One α] (i j : V) : A.compl i j = 0 ∨ A.compl i j = 1

@[simp]

theorem Matrix.isSymm_compl {V : Type u_1} {α : Type u_2} [DecidableEq α] [DecidableEq V] (A : Matrix V V α) [Zero α] [One α] (h : A.IsSymm) : A.compl.IsSymm

@[simp]

theorem Matrix.isAdjMatrix_compl {V : Type u_1} {α : Type u_2} [DecidableEq α] [DecidableEq V] (A : Matrix V V α) [Zero α] [One α] (h : A.IsSymm) : A.compl.IsAdjMatrix

@[simp]

theorem Matrix.IsAdjMatrix.compl {V : Type u_1} {α : Type u_2} [DecidableEq α] [DecidableEq V] {A : Matrix V V α} [Zero α] [One α] (h : A.IsAdjMatrix) : A.compl.IsAdjMatrix

theorem Matrix.IsAdjMatrix.toGraph_compl_eq {V : Type u_1} {α : Type u_2} [DecidableEq α] [DecidableEq V] {A : Matrix V V α} [MulZeroOneClass α] [Nontrivial α] (h : A.IsAdjMatrix) : ⋯.toGraph = h.toGraphᶜ

def SimpleGraph.adjMatrix {V : Type u_1} (α : Type u_2) (G : SimpleGraph V) [DecidableRel G.Adj] [Zero α] [One α] : Matrix V V α

adjMatrix G α is the matrix A such that A i j = (1 : α) if i and j are adjacent in the simple graph G, and otherwise A i j = 0.

Equations

• SimpleGraph.adjMatrix α G = Matrix.of fun (i j : V) => if G.Adj i j then 1 else 0

@[simp]

theorem SimpleGraph.adjMatrix_apply {V : Type u_1} {α : Type u_2} (G : SimpleGraph V) [DecidableRel G.Adj] (v w : V) [Zero α] [One α] : adjMatrix α G v w = if G.Adj v w then 1 else 0

@[simp]

theorem SimpleGraph.transpose_adjMatrix {V : Type u_1} {α : Type u_2} (G : SimpleGraph V) [DecidableRel G.Adj] [Zero α] [One α] : (adjMatrix α G).transpose = adjMatrix α G

@[simp]

theorem SimpleGraph.isSymm_adjMatrix {V : Type u_1} {α : Type u_2} (G : SimpleGraph V) [DecidableRel G.Adj] [Zero α] [One α] : (adjMatrix α G).IsSymm

@[simp]

theorem SimpleGraph.isAdjMatrix_adjMatrix {V : Type u_1} (α : Type u_2) (G : SimpleGraph V) [DecidableRel G.Adj] [Zero α] [One α] : (adjMatrix α G).IsAdjMatrix

The adjacency matrix of G is an adjacency matrix.

theorem SimpleGraph.toGraph_adjMatrix_eq {V : Type u_1} (α : Type u_2) (G : SimpleGraph V) [DecidableRel G.Adj] [MulZeroOneClass α] [Nontrivial α] : ⋯.toGraph = G

The graph induced by the adjacency matrix of G is G itself.

theorem SimpleGraph.one_add_adjMatrix_add_compl_adjMatrix_eq_allOnes {V : Type u_1} {α : Type u_2} (G : SimpleGraph V) [DecidableRel G.Adj] [DecidableEq V] [DecidableEq α] [AddMonoidWithOne α] : 1 + adjMatrix α G + (adjMatrix α G).compl = Matrix.of fun (x x : V) => 1

The sum of the identity, the adjacency matrix, and its complement is the all-ones matrix.

@[simp]

theorem SimpleGraph.adjMatrix_dotProduct {V : Type u_1} {α : Type u_2} (G : SimpleGraph V) [DecidableRel G.Adj] [Fintype V] [NonAssocSemiring α] (v : V) (vec : V → α) : adjMatrix α G v ⬝ᵥ vec = ∑ u ∈ G.neighborFinset v, vec u

@[simp]

theorem SimpleGraph.dotProduct_adjMatrix {V : Type u_1} {α : Type u_2} (G : SimpleGraph V) [DecidableRel G.Adj] [Fintype V] [NonAssocSemiring α] (v : V) (vec : V → α) : vec ⬝ᵥ adjMatrix α G v = ∑ u ∈ G.neighborFinset v, vec u

@[simp]

theorem SimpleGraph.adjMatrix_mulVec_apply {V : Type u_1} {α : Type u_2} (G : SimpleGraph V) [DecidableRel G.Adj] [Fintype V] [NonAssocSemiring α] (v : V) (vec : V → α) : (adjMatrix α G).mulVec vec v = ∑ u ∈ G.neighborFinset v, vec u

@[simp]

theorem SimpleGraph.adjMatrix_vecMul_apply {V : Type u_1} {α : Type u_2} (G : SimpleGraph V) [DecidableRel G.Adj] [Fintype V] [NonAssocSemiring α] (v : V) (vec : V → α) : Matrix.vecMul vec (adjMatrix α G) v = ∑ u ∈ G.neighborFinset v, vec u

@[simp]

theorem SimpleGraph.adjMatrix_mul_apply {V : Type u_1} {α : Type u_2} (G : SimpleGraph V) [DecidableRel G.Adj] [Fintype V] [NonAssocSemiring α] (M : Matrix V V α) (v w : V) : (adjMatrix α G * M) v w = ∑ u ∈ G.neighborFinset v, M u w

@[simp]

theorem SimpleGraph.mul_adjMatrix_apply {V : Type u_1} {α : Type u_2} (G : SimpleGraph V) [DecidableRel G.Adj] [Fintype V] [NonAssocSemiring α] (M : Matrix V V α) (v w : V) : (M * adjMatrix α G) v w = ∑ u ∈ G.neighborFinset w, M v u

@[simp]

theorem SimpleGraph.trace_adjMatrix {V : Type u_1} (α : Type u_2) (G : SimpleGraph V) [DecidableRel G.Adj] [Fintype V] [AddCommMonoid α] [One α] : (adjMatrix α G).trace = 0

theorem SimpleGraph.adjMatrix_mul_self_apply_self {V : Type u_1} {α : Type u_2} (G : SimpleGraph V) [DecidableRel G.Adj] [Fintype V] [NonAssocSemiring α] (i : V) : (adjMatrix α G * adjMatrix α G) i i = ↑(G.degree i)

theorem SimpleGraph.adjMatrix_mulVec_const_apply {V : Type u_1} {α : Type u_2} {G : SimpleGraph V} [DecidableRel G.Adj] [Fintype V] [NonAssocSemiring α] {a : α} {v : V} : (adjMatrix α G).mulVec (Function.const V a) v = ↑(G.degree v) * a

theorem SimpleGraph.adjMatrix_mulVec_const_apply_of_regular {V : Type u_1} {α : Type u_2} {G : SimpleGraph V} [DecidableRel G.Adj] [Fintype V] [NonAssocSemiring α] {d : ℕ} {a : α} (hd : G.IsRegularOfDegree d) {v : V} : (adjMatrix α G).mulVec (Function.const V a) v = ↑d * a

theorem SimpleGraph.adjMatrix_pow_apply_eq_card_walk {V : Type u_1} {α : Type u_2} {G : SimpleGraph V} [DecidableRel G.Adj] [Fintype V] [DecidableEq V] [Semiring α] (n : ℕ) (u v : V) : (adjMatrix α G ^ n) u v = ↑(Fintype.card ↑{p : G.Walk u v | p.length = n})

theorem SimpleGraph.dotProduct_mulVec_adjMatrix {V : Type u_1} {α : Type u_2} {G : SimpleGraph V} [DecidableRel G.Adj] [Fintype V] [NonAssocSemiring α] (x y : V → α) : x ⬝ᵥ (adjMatrix α G).mulVec y = ∑ i : V, ∑ j : V, if G.Adj i j then x i * y j else 0

theorem Matrix.IsAdjMatrix.adjMatrix_toGraph_eq {V : Type u_1} {α : Type u_2} [MulZeroOneClass α] [Nontrivial α] {A : Matrix V V α} (h : A.IsAdjMatrix) [DecidableEq α] : SimpleGraph.adjMatrix α h.toGraph = A

If A is qualified as an adjacency matrix, then the adjacency matrix of the graph induced by A is itself.