Degree (graph theory) From Wikipedia, the free encyclopedia Contents 1 Degree (graph theory) 1 1.1 Handshaking lemma .......................................... 1 1.2 Degree sequence ............................................ 2 1.3 Special values ............................................. 3 1.4 Global properties ........................................... 3 1.5 See also ................................................ 4 1.6 Notes ................................................. 4 1.7 References ............................................... 4 2 Graph operations 5 2.1 Unary operations ........................................... 5 2.1.1 Elementary operations .................................... 5 2.1.2 Advanced operations ..................................... 5 2.2 Binary operations ........................................... 5 2.3 Notes ................................................. 6 3 Regular graph 7 3.1 Existence ............................................... 7 3.2 Algebraic properties .......................................... 7 3.3 Generation .............................................. 8 3.4 See also ................................................ 8 3.5 References .............................................. 8 3.6 External links ............................................. 8 3.7 Text and image sources, contributors, and licenses .......................... 9 3.7.1 Text .............................................. 9 3.7.2 Images ............................................ 9 3.7.3 Content license ........................................ 9 i Chapter 1 Degree (graph theory) A graph with vertices labeled by degree In graph theory, the degree (or valency) of a vertex of a graph is the number of edges incident to the vertex, with loops counted twice.[1] The degree of a vertex v is denoted deg(v) or deg v . The maximum degree of a graph G, denoted by Δ(G), and the minimum degree of a graph, denoted by δ(G), are the maximum and minimum degree of its vertices. In the graph on the right, the maximum degree is 5 and the minimum degree is 0. In a regular graph, all degrees are the same, and so we can speak of the degree of the graph. 1.1 Handshaking lemma Main article: handshaking lemma The degree sum formula states that, given a graph G = (V; E) , X deg(v) = 2jEj : v2V 1 2 CHAPTER 1. DEGREE (GRAPH THEORY) The formula implies that in any graph, the number of vertices with odd degree is even. This statement (as well as the degree sum formula) is known as the handshaking lemma. The latter name comes from a popular mathematical problem, to prove that in any group of people the number of people who have shaken hands with an odd number of other people from the group is even. 1.2 Degree sequence Two non-isomorphic graphs with the same degree sequence (3, 2, 2, 2, 2, 1, 1, 1). The degree sequence of an undirected graph is the non-increasing sequence of its vertex degrees;[2] for the above graph it is (5, 3, 3, 2, 2, 1, 0). The degree sequence is a graph invariant so isomorphic graphs have the same degree sequence. However, the degree sequence does not, in general, uniquely identify a graph; in some cases, non-isomorphic graphs have the same degree sequence. The degree sequence problem is the problem of finding some or all graphs with the degree sequence being a given non-increasing sequence of positive integers. (Trailing zeroes may be ignored since they are trivially realized by adding an appropriate number of isolated vertices to the graph.) A sequence which is the degree sequence of some graph, i.e. for which the degree sequence problem has a solution, is called a graphic or graphical sequence. As a consequence of the degree sum formula, any sequence with an odd sum, such as (3, 3, 1), cannot be realized as the degree sequence of a graph. The converse is also true: if a sequence has an even sum, it is the degree sequence 1.3. SPECIAL VALUES 3 of a multigraph. The construction of such a graph is straightforward: connect vertices with odd degrees in pairs by a matching, and fill out the remaining even degree counts by self-loops. The question of whether a given degree sequence can be realized by a simple graph is more challenging. This problem is also called graph realization problem and can either be solved by the Erdős–Gallai theorem or the Havel–Hakimi algorithm. The problem of finding or estimating the number of graphs with a given degree sequence is a problem from the field of graph enumeration. 1.3 Special values An undirected graph with leaf nodes 4, 5, 6, 7, 10, 11, and 12 • A vertex with degree 0 is called an isolated vertex. • A vertex with degree 1 is called a leaf vertex or end vertex, and the edge incident with that vertex is called a pendant edge. In the graph on the right, {3,5} is a pendant edge. This terminology is common in the study of trees in graph theory and especially trees as data structures. • A vertex with degree n − 1 in a graph on n vertices is called a dominating vertex. 1.4 Global properties • If each vertex of the graph has the same degree k the graph is called a k-regular graph and the graph itself is said to have degree k. Similarly, a bipartite graph in which every two vertices on the same side of the bipartition as each other have the same degree is called a biregular graph. • An undirected, connected graph has an Eulerian path if and only if it has either 0 or 2 vertices of odd degree. If it has 0 vertices of odd degree, the Eulerian path is an Eulerian circuit. • A directed graph is a pseudoforest if and only if every vertex has outdegree at most 1. A functional graph is a special case of a pseudoforest in which every vertex has outdegree exactly 1. 4 CHAPTER 1. DEGREE (GRAPH THEORY) • By Brooks’ theorem, any graph other than a clique or an odd cycle has chromatic number at most Δ, and by Vizing’s theorem any graph has chromatic index at most Δ + 1. • A k-degenerate graph is a graph in which each subgraph has a vertex of degree at most k. 1.5 See also • Indegree, outdegree for digraphs • Degree distribution • degree sequence for bipartite graphs 1.6 Notes [1] Diestel p.5 [2] Diestel p.278 1.7 References • Diestel, Reinhard (2005), Graph Theory (3rd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540- 26183-4. • Erdős, P.; Gallai, T. (1960), “Gráfok előírt fokszámú pontokkal” (PDF), Matematikai Lapok (in Hungarian) 11: 264–274. • Havel, Václav (1955), “A remark on the existence of finite graphs”, Časopis pro pěstování matematiky (in Czech) 80: 477–480 • Hakimi, S. L. (1962), “On realizability of a set of integers as degrees of the vertices of a linear graph. I”, Journal of the Society for Industrial and Applied Mathematics 10: 496–506, MR 0148049. • Sierksma, Gerard; Hoogeveen, Han (1991), “Seven criteria for integer sequences being graphic”, Journal of Graph Theory 15 (2): 223–231, doi:10.1002/jgt.3190150209, MR 1106533. Chapter 2 Graph operations Operations on graphs produce new graphs from old ones. They may be separated into the following major cate- gories. 2.1 Unary operations Unary operations create a new graph from the old one. 2.1.1 Elementary operations These are sometimes called “editing operations” on graphs. They create a new graph from the original one by a simple, local change, such as addition or deletion of a vertex or an edge, merging and splitting of vertices, edge contraction, etc. 2.1.2 Advanced operations • Transpose graph • Complement graph • Line graph • Graph minor • Power of graph: The k-th power Gk of a graph G is a supergraph formed by adding an edge between all pairs of vertices of G with distance at most k. The second power of a graph is also called its square. • Dual graph (only applies to a graph embedded in a surface) • Medial graph • Y-Δ transform • Mycielskian 2.2 Binary operations Binary operations create a new graph from two initial graphs G1(V1, E1) and G2(V2, E2): • The union of two graphs G = (VG ,EG) and H = (VH ,EH) is the union of their vertex and edge sets: G ∪ H = (VG ∪ VH,EG ∪ EH). When VG and VH are disjoint, their union is referred to as the disjoint union, and denoted G + H.[1] 5 6 CHAPTER 2. GRAPH OPERATIONS • Similarly to above, the intersection of two graphs G = (VG ,EG) and H = (VH ,EH) is G ∩ H = (VG ∩ VH, EG ∩ EH).[1] • The graph join (or complete join) of two graphs is their graph union with all the edges that connect the vertices of the first graph with the vertices of the second graph.[2] It is a commutative operation (for unlabelled graphs) • Graph products based on the Cartesian product of the vertex sets: • Cartesian product of graphs[2] It is a commutative and associative operation (for unlabelled graphs). • Lexicographic product of graphs (also called graph composition) [2] It is associative (for unlabelled graphs), but not commutative. • Strong product of graphs; It is commutative and associative (for unlabelled graphs). • Tensor product of graphs, also called direct product, categorical product, cardinal product, or Kro- necker product. It is a commutative and associative operation (for unlabelled graphs). • Zig-zag product of graphs [3] Let [N] denote the set of integers from 1 to N. It is supposed that k-regular graphs used in the definition below are k-edge colored, i.e., their edge sets are partitioned into k perfect matchings. For each color i and a vertex v let v[i] denote the neighbor of v along the edge colored with color i. Let G1 be a D1-regular graph on [N1] and let G2 be a D2-regular graph on [D1]. Then the zig-zag product H is a graph with vertex set [N1] × [D1], where for all n in [N1], d in [D1], and i,j, in [D2], the vertex (n, d) is connected to (n[d[i]], d[i][j]).