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In graph theory, a k-degenerate graph is an undirected graph in which every subgraph has at least one vertex of degree at most k: that is, some vertex in the subgraph touches k or fewer of the subgraph's edges. The degeneracy of a graph is the smallest value of k for which it is k-degenerate.
In chemical graph theory, the Wiener index (also Wiener number) introduced by Harry Wiener, is a topological index of a molecule, defined as the sum of the lengths of the shortest paths between all pairs of vertices in the chemical graph representing the non-hydrogen atoms in the molecule.
k-degenerate graphs have also been called k-inductive graphs. degree 1. The degree of a vertex in a graph is its number of incident edges. [2] The degree of a graph G (or its maximum degree) is the maximum of the degrees of its vertices, often denoted Δ(G); the minimum degree of G is the minimum of its vertex degrees, often denoted δ(G).
If G is an undirected graph, then the degeneracy of G is the minimum number p such that every subgraph of G contains a vertex of degree p or smaller. A graph with degeneracy p is called p-degenerate. Equivalently, a p-degenerate graph is a graph that can be reduced to the empty graph by repeatedly removing a vertex of degree p or smaller.
Claw-free graphs: Star K 1,3: Induced subgraph Definition Comparability graphs: Induced subgraph Triangle-free graphs: Triangle K 3: Induced subgraph Definition Planar graphs: K 5 and K 3,3: Homeomorphic subgraph Kuratowski's theorem: K 5 and K 3,3: Graph minor Wagner's theorem: Outerplanar graphs: K 4 and K 2,3: Graph minor Diestel (2000), [1 ...
The degenerate univariate distribution can be viewed as the limiting case of a continuous distribution whose variance goes to 0 causing the probability density function to be a delta function at k 0, with infinite height there but area equal to 1. [citation needed] The cumulative distribution function of the univariate degenerate distribution is:
In 2009, a graph built by Gábor Wiener and Makoto Araya became (with its 42 vertices) the smallest planar hypohamiltonian graph known. [ 9 ] [ 10 ] In their paper, Wiener and Araya conjectured their graph to be optimal arguing that its order ( 42 ) appears to be the answer to The Ultimate Question of Life, the Universe, and Everything from The ...
The Hosoya index is the first topological index recognized in chemical graph theory, and it is often referred to as "the" topological index. [6] Other examples include the Wiener index, Randić's molecular connectivity index, Balaban’s J index, [7] and the TAU descriptors.