"...a regular graph is a graph where each vertex has the same number of neighbors; i.e. every vertex has the same degree or valency."
Regular graphs have all vertices with the same degree, or number of edges incident to each vertex. They are often used in the study of symmetry and patterns.
Graph Theory Basics: This includes the definition of a graph, degree of a vertex, edge, path, cycle, and connectivity.
Regular Graph: A graph is called regular if all of its vertices have the same degree.
Valency of a Graph: The valency of a graph is the number of edges that connect to a vertex.
Properties of Regular Graphs: This includes the properties of regular graphs such as even number of vertices, existence of Hamiltonian cycles, and existence of perfect matchings.
Degree Sequences: The degree sequence of a graph is the sequence of degrees of its vertices listed in non-increasing order.
Regularity Criterion: A graph with n vertices is regular of degree k if and only if its degree sequence is (k, k, ..., k).
Planar Graphs: A planar graph is a graph that can be drawn on a plane without any edge crossings.
Euler's Formula: This formula relates the number of vertices, edges, and faces of a planar graph.
Dual Graphs: The dual of a planar graph is a graph obtained by replacing each face of the planar graph by a vertex.
Graph Coloring: Graph coloring is the assignment of colors to the vertices of a graph.
Regular Graphs and their Chromatic Number: The chromatic number of a graph is the minimum number of colors needed to color the vertices of a graph.
Regular Graphs and their Applications: Regular graphs have applications in computer networking, coding theory, and social network analysis.
Regular Graphs and their Symmetries: Regular graphs have symmetries that can be studied using group theory.
Regular Graphs and their Automorphisms: An automorphism of a graph is an isomorphism from the graph to itself.
Degree-Diameter Problem: The degree-diameter problem is concerned with finding the largest possible degree for a graph of a given diameter.
Null Graph: It is a graph with no vertices and no edges.
Complete Graph: It is a graph in which each vertex is connected to every other vertex in the graph.
Circular Graph: It is a graph in which vertices are arranged in a circle and each vertex is connected to its two adjacent vertices.
Regular Bipartite Graph: It is a graph in which vertices can be divided into two sets such that each vertex in one set is connected to every vertex in the other set.
Regular Tree: It is a tree graph in which every non-leaf node has the same number of children.
Regular Grid Graph: It is a graph in which the vertices form a regular grid and each vertex is connected to its neighboring vertices.
Regular Hypercube Graph: It is a graph in which the vertices are arranged in a hypercube and each vertex is connected to its neighboring vertices.
Regular Archimedean Graph: It is a graph in which each vertex has the same degree and is arranged in a regular pattern.
Regular Toroidal Graph: It is a graph in which the vertices form a torus and each vertex is connected to its neighboring vertices.
Regular Möbius Graph: It is a graph in which the vertices form a Möbius strip and each vertex is connected to its neighboring vertices.
"A regular directed graph must also satisfy the stronger condition that the indegree and outdegree of each internal vertex are equal to each other."
"A regular graph with vertices of degree k is called a k‑regular graph or regular graph of degree k."
"From the handshaking lemma, a regular graph contains an even number of vertices with odd degree."
"A 0-regular graph consists of disconnected vertices, a 1-regular graph consists of disconnected edges, and a 2-regular graph consists of a disjoint union of cycles and infinite chains."
"A 3-regular graph is known as a cubic graph."
"A strongly regular graph is a regular graph where every adjacent pair of vertices has the same number l of neighbors in common, and every non-adjacent pair of vertices has the same number n of neighbors in common."
"The smallest graphs that are regular but not strongly regular are the cycle graph and the circulant graph on 6 vertices."
"The complete graph Km is strongly regular for any m."
"A theorem by Nash-Williams says that every k‑regular graph on 2k + 1 vertices has a Hamiltonian cycle."
"The handshaking lemma states that the sum of the degrees of all vertices in a graph is twice the number of edges."
"A regular directed graph must satisfy the stronger condition that the indegree and outdegree of each internal vertex are equal."
"No, in a regular graph, each vertex has the same degree or valency."
"A regular graph containing an even number of vertices with odd degree allows for the possibility of traversing each edge once and returning to the starting vertex."
"Regular graphs of degree at most 2 are classified as 0-regular, 1-regular, and 2-regular graphs."
"Strongly regular graphs are characterized by the same number of neighbors that adjacent vertices share, as well as the same number of neighbors that non-adjacent vertices share."
"Yes, cycle graphs are one of the smallest graphs that are regular and strongly regular."
"A k‑regular graph on 2k + 1 vertices is guaranteed to have a Hamiltonian cycle."
"No, a regular directed graph is a specific type of regular graph that satisfies additional conditions regarding indegree and outdegree."
"No, as per the handshaking lemma, a regular graph must contain an even number of vertices with odd degrees."