Graph theory gets even cooler when we look at special types of graphs. Bipartite graphs split vertices into two groups, complete graphs connect everything, and regular graphs have uniform degrees. These structures pop up everywhere!

Understanding these special graphs is key to solving real-world problems. They model everything from job assignments to social networks. Plus, they're the building blocks for more complex graph concepts we'll tackle later.

Bipartite graphs

Definition and characteristics

• Bipartite graph divides vertices into two disjoint sets with edges connecting vertices between sets
• Two disjoint sets called partite sets or color classes
• Graph bipartite if and only if it contains no odd-length cycles
• Complete bipartite graph Km,n connects every vertex in one set to every vertex in other set
• Recognize bipartite graphs using two-coloring algorithm assigning different colors to adjacent vertices
• Bipartite graphs precisely the graphs colored using only two colors

Applications and related concepts

• Used in matching problems (assigning jobs to workers, students to dormitories)
• Closely related to graph coloring concept
• Model relationships between two distinct groups (buyers and sellers, actors and movies)
• Represent alternating structures (chemical compounds, scheduling problems)
• Analyze social networks (identifying communities, studying interactions between groups)
• Optimize resource allocation (assigning tasks to processors, matching supply with demand)

Complete graphs

Properties and structure

• Simple undirected graph connecting every pair of distinct vertices with unique edge
• Denoted as Kn for n vertices
• Contains $n(n-1)/2$ edges
• Every vertex has degree $n-1$
• Maximally connected with highest possible number of edges for n vertices
• Complement graph an empty graph (no edges)
• Clique number equal to number of vertices

Applications and significance

• Model all-to-all communication networks
• Solve Traveling Salesman Problem using heuristics
• Analyze worst-case scenarios in algorithm complexity
• Study social networks (complete subgraphs represent tightly connected groups)
• Crucial in extremal graph theory and Ramsey theory
• Benchmark for testing graph algorithms
• Represent fully connected neural networks in machine learning

Regular graphs

Definition and types

• Graph where every vertex has same degree
• k-regular graph has each vertex with degree k
• Degree of regularity denoted by r (common degree of all vertices)
• Complete graph Kn example of (n-1)-regular graph
• Cube graph Q3 example of 3-regular graph (cubic graph)
• Regular graphs of degree 2 precisely disjoint unions of cycles
• Handshaking lemma implies number of vertices in odd degree regular graph must be even

Properties and examples

• Vertex-transitive (all vertices "look the same")
• Often exhibit high symmetry and interesting structural properties
• Petersen graph famous example of 3-regular graph with 10 vertices
• Platonic solids correspond to regular graphs (tetrahedron, cube, octahedron, dodecahedron, icosahedron)
• Complete bipartite graph Kn,n example of n-regular graph
• Strongly regular graphs special class with additional constraints on neighbor connections
• Cage graphs smallest regular graphs with given degree and girth

Degree sequences vs Graph types

Degree sequences and graph properties

• Degree sequence lists degrees of vertices, typically in non-increasing order
• Bipartite graphs have constraints on degree sequences related to sum of degrees in each partite set
• Complete graphs have uniform degree sequence (all entries equal to n-1)
• Regular graphs have constant degree sequence (all entries equal to degree of regularity r)
• Erdős-Gallai theorem provides necessary and sufficient conditions for sequence to be degree sequence of simple graph
• Havel-Hakimi algorithm constructs graph with given degree sequence or determines if such graph exists
• Degree sequence provides insights into graph properties (connectivity, planarity, existence of Hamiltonian cycles)

Analyzing and constructing graphs from degree sequences

• Graphical sequences satisfy specific conditions (sum of degrees even, largest degree less than number of vertices)
• Degree sequence majorization compares sequences to determine potential graph structures
• Degree sequence partitions used to classify and analyze graph families
• Threshold graphs characterized by special properties of their degree sequences
• Degree sequence reconstruction problem attempts to determine unique graphs from degree sequences
• Degree sequence analysis helps in studying random graph models and their properties
• Degree sequence constraints used in graph generation algorithms and network modeling