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Graph Theory

Recommended Reading : 【Mathematics】 Mathematics Table of Contents


1. Basics

2. Applications



1. Basics

⑴ Components of a graph

① V: Node

② E: Edge

③ G: Graph

④ V(G) = {v1, ···, vn}

⑤ vi ~ vj: When edge eij connects vertices vi, vj.

⑵ Degree of Graph G : Number of nodes, i.e., V(G)
⑶ Size of Graph G : Number of edges, i.e., E(G)

⑷ Degree of node v : Number of edges connected to node v, i.e., deg(v).

① deg(vi) = ∑vi~vj 1

② δ(G) refers to the minimum degree of nodes in G, and Δ(G) refers to the maximum degree of nodes in G.

③ ∑ deg(vi) = 2 E(G)

④ If all the nodes of a graph have the same degree, the graph is regular.

⑤ The nodes of an Eulerian graph have even degree.

⑸ Simple graph and complete graph

① Simple graph : A graph in which at most one edge exists between any two nodes, and self-loops (edges connecting a node to itself) are not allowed.


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Figure 1. Simple Graph


② Complete graph : A graph in which all nodes are connected by unique edges.


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Figure 2. Complete Graph


○ strongly connected (= irreducible) : A state where any node i can reach any other node j within the graph.

○ The number of edges in a complete graph is NC2 = N(N-1) / 2.

⑹ Subgraph: H is a subgraph of G if V(H) ⊆ V(G) and E(H) ⊆ E(G).

① A subgraph H is an induced subgraph of G if two vertices of V(H) are adjacent if and only if they are adjacent in G.

② clique: a complete subgraph of a graph.

③ Walk : A path like v0 → v1 → … → vm, connected by edges v0v1, v1v2, …, vm-1vm.

○ v0 is called the initial vertex, and vm is called the final vertex.

○ The number of edges in the walk is called the length.

④ Path: A walk where all nodes are distinct.

⑤ Cycle: A walk where v0 = vm.

⑥ Forest: A graph containing no cycles.

⑦ Tree: A connected forest

⑧ Period : The period of a node i is the greatest common divisor of all paths from node i to itself.

○ Example : If two nodes A and B are connected by two edges such that A = B, the period of each node is 2.

⑨ Aperiodic : When the period of all nodes is 1.

○ aperiodic ⊂ irreduicible

○ Example : A walk where each node has a path leading back to itself is aperiodic.

⑩ Regular

○ regular ⊂ irreducible

○ For some natural number k, all elements of the k-th power of the transition matrix M, Mk, are positive (i.e., non-zero values).

⑺ Types of graph

① Undirected graph: A graph with undirected edges

② Directed graph: A graph with directed edges

③ k-partite graph: If its set of vertices admits a partition into k classes such that the vertices of the same class are not adjacent.

⑻ Adjacency matrix: Represented as A.

① Definition: When the number of nodes is n, an n × n adjacency matrix is defined as, Ai,j = 1 (if vi ~ vj), Ai,j = 0 (otherwise)


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Figure 3. Example of adjancy matrix


② Real-valued function on the set of the graph’s vertices, f: V → ℝ


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Figure 4. Example of the real-valued function f


③ The following fomula holds


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⑼ Incidence matrix: Represented as ∇.

① Incidence matrix of undirected graph is defined as, Bi,j = 1 (if vertex vi is incident with edge ej), Bi,j = 0 (otherwise)


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Figure 5. Example of incidence matrix


② Incidence matrix of directed graph


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⑽ Laplacian matrix

① Overview

○ Definition: L = ∇T∇ = D - A (cf. Dii = d(vi))

○ Multiplicity: # of connected components of the graph


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Figure 6. Example of Laplacian matrix


② Modification of Laplacian matrix

○ Normalized graph Laplacian: Ln = D-1/2LD-1/2 = I - D-1/2AD-1/2. symmetric, semi-definite positive

○ Transition matrix of Markov chain: Lt = D-1A

○ Random-walk graph Laplacian: Lr = D-1L = I - Lt = D-1/2LnD1/2

Characteristic 1. L has an eigenvalue of 0. L1n = 0

○ If there are k disconnected graphs, there are k eigenvalues of 0.

○ The closer the eigenvalue is to 0, the more connected the graph is.

○ Fiedler vector: The eigenvector corresponding to the second smallest eigenvalue after λ1 = 0. It represents the overall structure of the graph.

Characteristic 2. L is symmetric and positive semi-definite.

○ Quadratic form: fTLf = ∑ei,j, i < j (f(vi) - f(vj))2

○ That is, L has n non-negative, real-valued eigen values.

Application 1. Spectral Clustering

Step 1: Calculate the distance between the given n data points to create an n × n adjacency matrix A.

○ The Gaussian kernel is often used.


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Step 2: Based on the adjacency matrix, calculate the Laplacian matrix: Ln or Lr can also be used.

Step 3: Calculate the matrix of eigenvectors W = [ω2, ···, ωk] ∈ ℝn×k corresponding to the k smallest eigenvalues of the Laplacian matrix.

○ Exclude ω1 = 1, which corresponds to λ1 = 0, as it is trivial.

Step 4: The n data points are naturally embedded into k dimensions: Y = WT = [y1, ⋯, yn], with each column vector representing the embedded data points.

Step 5: Perform clustering, such as K-means clustering, on the column vectors of Y.

⑾ Degree distribution : The ratio of nodes with degree k among all nodes, i.e., P(k) = Nk / V(G) .

⑿ Distance : The length of the shortest path between two nodes. If no such path exists, the distance is defined as ∞.

⒀ Clustering coefficient : If the degree of a specific node is ki and the number of edges between the surrounding nodes connected to that node is ei, then:


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① For example, if A is connected to B, C, and D, and there are additional edges B-C and C-D, then for A, ei = 2, ki = 3, Ci = 2 / 3.

② ki (ki - 1) / 2 : The maximum number of edges that could be formed between the surrounding nodes.

③ Ci has a value between 0 and 1, with a value close to 1 indicating that the surrounding nodes are fully connected.



2. Applications

⑴ Pigeonhole principle

The number of unique graphs depending on the number of nodes : Nearly perfectly follows an exponential function.

Dijkstra and Floyd algorithms

Algorithm for counting the number of subgraphs

The power of belief

⑹ Social network theory or six degrees of separation

① The theory that any two people on Earth are connected by no more than six intermediaries.

② Proof : If one person knows an average of 100 people, then after six steps, 1006 = 1 billion, reaching a similar conclusion.

⑺ Perron-Frobenius theorem

Theorem 1. If the Markov chain with transition matrix P is strongly connected, then there is exactly one steady-state distribution q.

○ The steady-state distribution satisfies Pq = q.

Theorem 2. If it is also aperiodic, then:

○ Pij : The probability of transitioning from node j to node i. ∑i Pij = 1.

2-1. the matrix Pk converges to a matrix with columns all equal to q as k → ∞; i.e., Pij(k) converges to qi (for any i and j) as k → ∞

2-2. for any initial state x0 is, the Markov Chain {xk} converges to q as k → ∞

⑻ Google’s PageRank Algorithm

Step 1. Represent web pages as a graph and express it as a Markov process transition matrix P.

Step 2. For sink nodes, forcibly assign the transition probability to other pages as 1/n: P → P’.

Step 3. P’’ = (1 - α) P’ + (α / n) 1.

○ Here, 1 refers to an n × n matrix filled with ones.

○ While P’ is not guaranteed to be regular, P’’ is always guaranteed to be regular.

○ P’’ is a Markov chain defined by Google’s founders, Sergey Brin and Larry Page.

Step 4. Search for the steady-state vector x such that P’‘x = x.

○ Since P’’ is regular, it is strongly connected, which ensures the existence of a unique steady-state.

○ The state-transition probability at this point can be considered as the importance of the recommended page.

⑼ Four color theorem

① Any map can be colored with at most four colors, ensuring that no two adjacent regions share the same color.

② The theorem was proven by verifying that the four-color theorem holds in all cases using a computer.

⑽ Eulerian path problem or Königsberg bridge problem

① An Eulerian path, which passes through each node exactly once, exists in a graph if and only if there are 0 or 2 nodes with odd degrees.

② If there are 2 nodes with odd degrees, one of these nodes is the starting point and the other is the endpoint.

③ There is no path that crosses all seven bridges of Königsberg exactly once.

⑾ Ramsey theory

① In any sufficiently large graph, a certain structure (e.g., a complete graph of acquaintances or a subgraph of strangers) must exist.

② R(s, t) : The minimum number of nodes such that a subgraph of s strangers or a subgraph of t acquaintances is guaranteed to exist.

Theorem 1. R(3, 3) = 6 : Among any 6 people, there are either 3 mutual friends or 3 mutual strangers. This can be proven by counting possibilities.

Theorem 2. R(3, 3, 3) = 17.

Theorem 3. R(3, 3, 4) = 30.

Theorem 4. Recurrence relation of R(s, t) : R(s, t) ≤ R(s-1, t) + R(s, t-1).

Theorem 5. Existence of an upper bound for R(s, t) : Proven through the recurrence relation of R(s, t) and mathematical induction (ref).


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Theorem 6. Rk(s1, s2, …, sk) ≤ Rk/2(R(s1, s2), R(s3, s4), …, R(sk-1, sk)).

⑿ Schur’s theorem

① For any k ≥ 2, there exists an n > 3 such that under k-coloring, there always exists a (x, y, z) satisfying x + y = z, colored the same way in {1, 2, …, n}. Proven through Ramsey theory.

② This is connected to Fermat’s Last Theorem, revealing the existence of (x, y, z) where xm + ym = zm (mod p).

⒀ Erdős–Rényi Random Graph

① A very large graph emerges at a certain threshold, even with a simple rule of connecting nodes randomly.



Input: 2024.10.01 19:44

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