Graphs and Algebraic Matchings

• Vertices/Nodes (V): The set of entities (e.g., V = \{1, 2, 3, 4\}).
• Edges (E): The set of connections between vertices.
  • Undirected E \subseteq \{\{u, v\} \mid u, v \in V\}.
  • Directed E \subseteq V \times V.
• Adjacency: Two vertices u, v are adjacent if \{u, v\} \in E.
• Path-connectivity: A graph is connected if there is a path between every pair of vertices.

• Degree d(v): The number of edges incident to vertex v.
  • For directed graphs:
    • in-degree d^-(v): number of incoming edges.
    • out-degree d^+(v): number of outgoing edges.
• Handshaking Lemma: The sum of degrees in any undirected graph is exactly twice the number of edges: \sum_{v \in V} d(v) = 2|E|
  • Proof Intuition: Every edge \{u, v\} has two endpoints, contributing +1 to d(u) and +1 to d(v). Thus, each edge is counted exactly twice.
• Corollary: Every graph contains an even number of vertices with odd degree.

• Subgraph H = (V', E'): A graph where V' \subseteq V and E' \subseteq E.
  • An induced subgraph G[V'] keeps all edges from G that connect vertices in V'.
• Edge Contraction (G / e): Merging the endpoints of an edge e = (u, v).
  • Vertices u and v are replaced by a single new vertex uv.
  • All edges incident to u or v now connect to uv.
  • Important: Self-loops are deleted; parallel edges are kept (creating a multigraph).
  • Contraction reduces the vertex count by exactly 1.

• Path (P_n): A sequence of vertices connected line-by-line: v_1 - v_2 - \dots - v_n.
• Cycle (C_n): A path loop returning to its start: v_1 - v_2 - \dots - v_n - v_1.
• Tree: A connected acyclic graph (no cycles). Has exactly n-1 edges.
• Complete Graph (K_n): Every pair of distinct vertices is connected by a unique edge. Has exactly \binom{n}{2} edges.

• Bipartite Graph: Vertices can be split into two sets U and W such that every edge has one endpoint in U and one in W. No edges exist within U or within W.
• Hamiltonian Cycle: A closed loop (cycle) that visits every vertex in the graph exactly once.
• Planar Graph: A graph that can be drawn in the plane such that no two edges intersect except at their endpoints.
  • Example: K_4 is planar (can be drawn without crossings), but the complete graph K_5 is non-planar.

• The Web Graph: Modeling the entire web as a directed graph G = (V, E), where pages are vertices and hyperlinks are directed edges.
• The Algorithm (Random Surfer):
  • A user browses the web. At page u, they follow a random out-link with probability 1 - d.
  • With probability d (damping factor, typically 0.15), they teleport to a completely random page.
• PageRank Score: Over time, the probability of being at page v converges to its PageRank score: P(v) = \frac{d}{n} + (1-d) \sum_{u \in \text{In}(v)} \frac{P(u)}{d^+(u)}.
• Spectral Interpretation: This is the stationary distribution of a random walk (finding the principal eigenvector of the Google matrix).

• Seat Allocation (Stable Matching): Match candidates to college seats based on preferences (e.g., JoSAA counselling) such that no candidate and college mutually prefer each other over their assigned match.
• Network Reliability (Global Min-Cut): Find the minimum set of links to disconnect a network or separate an image, revealing the structural bottleneck.
• Route Planning (Travelling Salesperson): Find the minimum-cost closed tour (Hamiltonian cycle) that visits every city exactly once.

• Matching (M): A subset of edges M \subseteq E such that no two edges share a common vertex.
  • Vertices incident to an edge in M are matched; others are unmatched.
• Maximal Matching: A matching that cannot be expanded by adding another edge (no more independent edges can fit).
• Maximum Matching: A matching containing the largest possible number of edges.
• Perfect Matching: A matching of size exactly n/2 (every single vertex in the graph is matched).
  • Condition: A perfect matching can only exist if n is even.
• Stable Matching (Teaser): Pair elements from two sets based on preferences such that no two elements prefer each other over their assigned partners.

• Cut (S, V \setminus S): A partition of the vertex set V into two disjoint, non-empty subsets S and V \setminus S.
• Cut-Set: The set of edges with one endpoint in S and the other in V \setminus S.
• Size/Capacity of a Cut:
  • Unweighted: The number of crossing edges.
  • Weighted: The sum of weights of the crossing edges.
• Notation: c(S, V \setminus S) = |\{ (u, v) \in E \mid u \in S, v \notin S \}|.
• Goal (Min-Cut): Find a cut (S, V \setminus S) that minimizes this crossing size (the “bottleneck” of the graph).

• Lovász’s Theorem (1979):
  • Create a Tutte Matrix T where we put a variable x_{ij} if edge (i,j) exists (and -x_{ji} symmetrically).
  • The graph has a perfect matching \iff \det(T) is not the zero polynomial.
• The Problem:
  • Expanding \det(T) symbolically is exponential (n! terms).
  • But checking if \det(T) \not\equiv 0 is exactly a PIT instance!

Tutte Matrix T for K_4:
[  0   x12  x13  x14 ]
[-x12   0   x23  x24 ]
[-x13 -x23   0   x34 ]
[-x14 -x24 -x34   0  ]

det(T) = (x12*x34 - x13*x24 + x14*x23)^2
This is NOT identically zero!
So K_4 has a perfect matching.