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A vertex with degree 1 is called a leaf vertex or end vertex or a pendant 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 ...
The degree or valency of a vertex is the number of edges that are incident to it, where a loop is counted twice. The degree of a graph is the maximum of the degrees of its vertices. In an undirected simple graph of order n, the maximum degree of each vertex is n − 1 and the maximum size of the graph is n(n − 1) / 2 .
The degree or valency of a vertex is the number of edges that are incident to it; for graphs with loops, a loop is counted twice. In a graph of order n, the maximum degree of each vertex is n − 1 (or n + 1 if loops are allowed, because a loop contributes 2 to the degree), and the maximum number of edges is n(n − 1)/2 (or n(n + 1)/2 if loops ...
In graph theory, 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. A regular directed graph must also satisfy the stronger condition that the indegree and outdegree of each internal vertex are equal to each other. [1]
An edge coloring of a graph is a proper coloring of the edges, meaning an assignment of colors to edges so that no vertex is incident to two edges of the same color. An edge coloring with k colors is called a k-edge-coloring and is equivalent to the problem of partitioning the edge set into k matchings.
Similarly, an external vertex (or outer vertex, terminal vertex or leaf) is a vertex of degree 1. A branch vertex in a tree is a vertex of degree at least 3. [19] An irreducible tree (or series-reduced tree) is a tree in which there is no vertex of degree 2 (enumerated at sequence A000014 in the OEIS). [20]
The edge-connectivity for a graph with at least 2 vertices is less than or equal to the minimum degree of the graph because removing all the edges that are incident to a vertex of minimum degree will disconnect that vertex from the rest of the graph. [1] For a vertex-transitive graph of degree d, we have: 2(d + 1)/3 ≤ κ(G) ≤ λ(G) = d. [11]
One of the two vertices joined by a given edge, or one of the first or last vertex of a walk, trail or path. The first endpoint of a given directed edge is called the tail and the second endpoint is called the head. enumeration Graph enumeration is the problem of counting the graphs in a given class of graphs, as a function of their order. More ...