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• Graphs, Matrices and Generalized Inverse

Dr. Manjunatha Prasad Karantha [email protected], [email protected]

Professor of Mathematics Department of Statistics, PSPH, MAHE, Manipal

Coordinator Center for Advanced Research in Applied Mathematics and Statistics (CARAMS), MAHE, Manipal

December 10-21, 2018

• Contents

1 Introduction

2 Some Properties of Incidence Matrices

3 Minors of incidence matrix

4 Moore–Penrose Inverse of Incidence Matrix

5 Properties of 0-1 Incidence Matrix

6 Generalized inverse of Laplacian Matrix

7 More results on generalized inverses and graphs

8 References

K. Manjunatha Prasad (MAHE, Manipal) G-inverses & Graphs 06 December, 2018 2 / 50

• Introduction

Notationsℝ : set of real numbers ℝ𝑚×𝑛 : set of real matrices of size 𝑚 × 𝑛 ℝ𝑛 : set of real column vectors of order 𝑛 𝟏 : (1,… , 1)𝑇 {𝟏} : span of 𝟏

For 𝐴 ∈ ℝ𝑚×𝑛, 𝐴𝑇 : transpose of 𝐴 𝐴∗ : conjucate transpose of 𝐴(𝐴) : column space of 𝐴(𝐴) : (𝐴𝑇 ) rank(𝐴) : rank of the matrix 𝐴 𝐴𝑖 : 𝑖th row of 𝐴 𝐴𝑗 : 𝑗th column of 𝐴

For any two subspaces 𝑆, 𝑇 of ℝ𝑛, 𝑆 ⊕ 𝑇 : 𝑆 + 𝑇 with 𝑆 ∩ 𝑇 = {0} 𝑆 ⟂ 𝑇 : 𝑆 and 𝑇 are orthogonal to each other 𝑆 ⦹ 𝑇 : 𝑆 + 𝑇 with 𝑆 ⟂ 𝑇

K. Manjunatha Prasad (MAHE, Manipal) G-inverses & Graphs 06 December, 2018 3 / 50

• Introduction

Preliminaries

Given an 𝑚 × 𝑛 matrix 𝐴 over a real field, consider the following matrix equations, called Penrose Equations;

(1) 𝐴𝑋𝐴 = 𝐴 (2) 𝑋𝐴𝑋 = 𝑋 (3) (𝐴𝑋)𝑇 = 𝐴𝑋 (4) (𝑋𝐴)𝑇 = 𝑋𝐴.

A matrix 𝐺 satisfying the condition(s) ∙ (1) → generalized inverse of 𝐴, denoted by 𝐴−

∙ (2) → outer inverse of 𝐴, denoted by 𝐴=

∙ (1) − (4) → Moore-Penrose inverse of 𝐴, denoted by 𝐴+

{𝐴−} = {𝐺 + (𝐼 − 𝐺𝐴)𝑈 + 𝑉 (𝐼 − 𝐴𝐺) ∶ 𝐺 is any g-inverse of 𝐴, 𝑈 and 𝑉 are arbitrary matrices} (1)

K. Manjunatha Prasad (MAHE, Manipal) G-inverses & Graphs 06 December, 2018 4 / 50

• Introduction

Preliminaries

Definition (Space Equivalence) For any two matrices 𝐴 and 𝐵,

∙ 𝐴 ≃ 𝐶𝑆

𝐵 if (𝐴) = (𝐵) ∙ 𝐴 ≃

𝑅𝑆 𝐵 if (𝐴) = (𝐵)

∙ 𝐴 ≃ 𝑆𝑃

𝐵 if (𝐴) = (𝐵) and (𝐴) = (𝐵) Lemma For any matrix 𝐴, 𝐴+ ≃

𝑆𝑃 𝐴𝑇 .

K. Manjunatha Prasad (MAHE, Manipal) G-inverses & Graphs 06 December, 2018 5 / 50

• Introduction

Preliminaries

Lemma For given non-null matrices 𝐴, 𝐵 and 𝐶 in ℝ𝑚×𝑛, the following are equivalent.

(i) 𝐵𝐴−𝐶 is invariant under the choices of 𝐴−

(ii) (𝐶) ⊆ (𝐴) and (𝐵) ⊆ (𝐴). Sketch of the proof. (𝑖) ⟹ (𝑖𝑖). Let 𝐺 ∈ {𝐴−}. For 𝐴− = 𝐺 + (𝐼 −𝐺𝐴)𝑈 + 𝑉 (𝐼 − 𝐴𝐺), where 𝑈, 𝑉 are arbitrary, 𝐵𝐴−𝐶 = 𝐵𝐺𝐶 implies that 𝐵(𝐼 − 𝐺𝐴)𝑈𝐶 = 0 = 𝐵𝑉 (𝐼 − 𝐴𝐺)𝐶 . Hence, 𝐵(𝐼 − 𝐺𝐴) = 0 and (𝐼 − 𝐴𝐺)𝐶 = 0, proving (𝑖𝑖). (𝑖𝑖) ⟹ (𝑖). If 𝐶 = 𝐴𝑋 and 𝐵 = 𝑌 𝐴 for some 𝑋 and 𝑌 , then

𝐵𝐴−𝐶 = 𝑌 𝐴𝐴−𝐴𝑋 = 𝑌 𝐴𝑋,

independent of 𝐴−. □ K. Manjunatha Prasad (MAHE, Manipal) G-inverses & Graphs 06 December, 2018 6 / 50

• Introduction

Some matrices associated with graphs

Graph

Incidence Matrix

Laplacian Matrix

Distance Matrix

K. Manjunatha Prasad (MAHE, Manipal) G-inverses & Graphs 06 December, 2018 7 / 50

• Introduction

Incidence Matrix

Let 𝐺 be a graph with 𝑉 (𝐺) = {1,… , 𝑛} and 𝐸(𝐺) = {𝑒1,… , 𝑒𝑚}. For any directed graph 𝐺, the incidence matrix, denoted by 𝑄(𝐺), is the 𝑛 × 𝑚 matrix defined as

(𝑄(𝐺))𝑖𝑗 =

⎧⎪⎨⎪⎩ 0 vertex 𝑖 and edge 𝑒𝑗 are not incident 1 𝑒𝑗 originates at 𝑖 −1 𝑒𝑗 terminates at 𝑖

.

The incidence matrix of the graph in Figure 1.1 is,

𝑄 =

⎛⎜⎜⎜⎜⎝

𝑒1 𝑒2 𝑒3 𝑒4 𝑒5 𝑒6 1 −1 1 −1 0 0 0 2 1 0 0 −1 0 0 3 0 −1 0 0 1 0 4 0 0 1 0 0 −1 5 0 0 0 1 −1 1

⎞⎟⎟⎟⎟⎠

1

2 3 4

5

𝑒2𝑒1

𝑒5

𝑒3

𝑒4 𝑒6

Figure 1.1: Directed Graph

K. Manjunatha Prasad (MAHE, Manipal) G-inverses & Graphs 06 December, 2018 8 / 50

• Introduction

Some important facts about 𝑄(𝐺)

For any graph 𝐺, ∙ Always the column sum of 𝑄(𝐺) is zero.

Therefore, the rows are dependent. ∙ Rank of 𝑄(𝐺) ≤ 𝑛 − 1. ∙ Consider any left null vector 𝑥 of 𝑄(𝐺). Then,

𝑥𝑇𝑄(𝐺) = 0 ⟹ 𝑥𝑖 − 𝑥𝑗 = 0 if 𝑖 ∼ 𝑗. ⟹ 𝑥𝑖 = 𝑥𝑗∀𝑖, 𝑗, if 𝐺 is connected

⎛⎜⎜⎜⎜⎝

𝑒1 𝑒2 𝑒3 𝑒4 1 −1 1 −1 0 2 1 0 0 −1 3 0 −1 0 0 4 0 0 1 0 5 0 0 0 1

⎞⎟⎟⎟⎟⎠ Lemma

If 𝐺 is a connected graph on 𝑛 vertices, then rank 𝑄(𝐺) = 𝑛 − 1.

K. Manjunatha Prasad (MAHE, Manipal) G-inverses & Graphs 06 December, 2018 9 / 50

• Introduction

Rank of 𝑄(𝐺)

Lemma For any directed graph 𝐺 with 𝑘 components, the rank of 𝑄(𝐺) = 𝑛 − 𝑘.

Sketch of the proof. Let 𝐺1,… , 𝐺𝑘 be the connected components of 𝐺. Then, after a relabeling of vertices(rows) and edges (columns) if necessary, we have

𝑄(𝐺) =

⎡⎢⎢⎢⎣ 𝑄(𝐺1) 0 … 0

0 𝑄(𝐺2) … 0 ⋮ ⋮ ⋱ ⋮ 0 0 … 𝑄(𝐺𝑘)

⎤⎥⎥⎥⎦ .

K. Manjunatha Prasad (MAHE, Manipal) G-inverses & Graphs 06 December, 2018 10 / 50

• Introduction

Rank of 𝑄(𝐺) (contd.)

Since 𝐺𝑖 is connected, rank 𝑄(𝐺𝑖) is 𝑛𝑖 − 1, where 𝑛𝑖 is the number of vertices in 𝐺𝑖, 𝑖 = 1,… , 𝑘. It follows that

rank 𝑄(𝐺) = rank 𝑄(𝐺1) +⋯ + rank 𝑄(𝐺𝑘) = (𝑛1 − 1) +⋯ + (𝑛𝑘 − 1) = 𝑛1 +⋯ + 𝑛𝑘 = 𝑛 − 𝑘

This completes the proof. □

K. Manjunatha Prasad (MAHE, Manipal) G-inverses & Graphs 06 December, 2018 11 / 50

• Introduction

Let 𝐺 be a graph with 𝑉 (𝐺) = {1,… , 𝑛} and 𝐸(𝐺) = {𝑒1,… , 𝑒𝑚}. The adjacency matrix of 𝐺, denoted by 𝐴(𝐺), is the 𝑛 × 𝑛 matrix defined as follows.

(𝐴(𝐺))𝑖𝑗 =

{ 1 if vertex 𝑖 and vertex 𝑗 are adjacent 0 otherwise

.

1

2 3 4

5

𝑒2𝑒1

𝑒5

𝑒3

𝑒4 𝑒6 𝐴(𝐺) =

⎛⎜⎜⎜⎜⎜⎝

1 2 3 4 5 1 0 1 1 1 0 2 1 0 0 0 1 3 1 0 0 0 1 4 1 0 0 0 1 5 0 1 1 1 0

⎞⎟⎟⎟⎟⎟⎠ K. Manjunatha Prasad (MAHE, Manipal) G-inverses & Graphs 06 December, 2018 12 / 50

• Introduction

0–1 Incidence Matrix

Let 𝐺 be a graph with 𝑉 (𝐺) = {1,… , 𝑛} and 𝐸(𝐺) = {𝑒1,… , 𝑒𝑚}. The (vertex-edge) incidence matrix of 𝐺, which we denote by 𝑀(𝐺), or simply by 𝑀 , is the 𝑛 × 𝑚 matrix defined as follows.

(𝑀(𝐺))𝑖𝑗 =

{ 0 vertex 𝑖 and edge 𝑒𝑗 are not incident 1 otherwise.

The 0–1 incidence matrix of the graph in Figure 1.2 is,

𝑀(𝐺) =

⎛⎜⎜⎜⎜⎝

𝑒1 𝑒2 𝑒3 𝑒4 𝑒5 𝑒6 1 1 1 1 0 0 0 2 1 0 0 1 0 0 3 0 1 0 0 1 0 4 0 0 1 0 0 1 5 0 0 0 1 1 1

⎞⎟⎟⎟⎟⎠

1

2 3 4

5

𝑒2𝑒1

𝑒5

𝑒3

𝑒4 𝑒6

Figure 1.2: Directed GraphK. Manjunatha Prasad (MAHE, Manipal) G-inverses & Graphs 06 December, 2018 13 / 50

• Introduction

Degree Matrix

The degree matrix of a graph is a diagonal matrix with the vertex degrees on its diag- onal.

𝐷𝑒𝑔(𝐺) =

⎛⎜⎜⎜⎜⎝

1 2 3 4 5 1 3 0 0 0 0 2 0 2 0 0 0 3 0 0 2 0 0 4 0 0 0 2 0 5 0 0 0 0 3

⎞⎟⎟⎟⎟⎠

1

2 3 4

5

𝑒2𝑒1

𝑒5

𝑒3

𝑒4 𝑒6

K. Manjunatha Prasad (MAHE, Manipal) G-inverses & Graphs 06 December, 2018 14 / 50

• Introduction

Laplacian Matrix

The laplacian matrix of a given graph 𝐺 is, denoted by 𝐿(𝐺), is given by

(𝐿(𝐺))𝑖𝑗 =

⎧⎪⎨⎪⎩ −1 𝑖 ≠ 𝑗, vertex 𝑖 and vertex 𝑗 are adjacent 0 𝑖 ≠ 𝑗 vertex 𝑖 and vertex 𝑗 are not adjacent 𝑑𝑖 i=j

.

𝐿(𝐺) =

⎛⎜⎜⎜⎜⎜⎜⎝

1 2 3 4 5 6 1 3 −1 0 −1 −1 0 2 −1 3 −1 0 −1 0 3 0 −1 2 0 −1 0 4 −1 0 0 2 −1 0 5 −1 −1 −1 −1 5 −1 6 0 0 0 0 −1 1

⎞⎟⎟⎟⎟⎟⎟⎠ .

∙1 ∙2 ∙3

∙4 ∙5 ∙6

K. Manjunatha Prasad (MAHE, Manipal) G-inverses & Graphs 06 December, 2018 15 / 50

• Introduction

Laplacian Matrix (contd.)

Fac