Abstract
Average degree of a graph is defined to be a graph invariant equal to the arithmetic mean of all vertex degrees and has many applications, especially in determining the irregularity degrees of networks and social sciences. In this study, some properties of average degree have been studied. Effect of vertex deletion on this degree has been determined and a new proof of the handshaking lemma has been given. Using a recently defined graph index called o m e g a index, average degree of trees, unicyclic, bicyclic, and tricyclic graphs have been given, and these have been generalized to k -cyclic graphs. Also, the effect of edge deletion has been calculated. The average degree of some derived graphs and some graph operations have been determined.
Highlights
Let G (V, E) be a finite, undirected, and simple graph, having |V| n vertices and |E| m edges
We make use of Ω index to study the properties of average degree dG of a graph G which measures how many edges has G compared to the number of vertices, that is, dG
Using the obtained formula successively, we give a result which helps to calculate the average degree of a large graph by means of the average degree of a much smaller graph
Summary
Let G (V, E) be a finite, undirected, and simple graph, having |V| n vertices and |E| m edges. We make use of Ω index to study the properties of average degree dG of a graph G which measures how many edges has G compared to the number of vertices, that is, dG. Using the Ω index, graphs are classified according to their cyclicness, and for each case, the density of the graph is characterized as follows:. E average vertex degree of a connected k-cyclic graph is. As a result, connected acyclic, unicyclic, bicyclic, tricyclic, etc., graphs have average degrees 2(n − 1)/n, 2, 2(n + 1)/n, 2(n + 2)/n, etc., respectively
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