Abstract
In 2012, Abdo and Dimitrov defined the total irregularity of a graphG=(V,E)asirrtG=1/2∑u,v∈VdGu-dGv, wheredGudenotes the vertex degree of a vertexu∈V. In this paper, we investigate the total irregularity of bicyclic graphs and characterize the graph with the maximal total irregularity among all bicyclic graphs onnvertices.
Highlights
Let G = (V, E) be a simple undirected graph with vertex set V and edge set E
V ∈ V, the degree of V is denoted by dG(V), and the distance dG(u, V) is defined as the length of the shortest path between u and V in G
In [1], Albertson defined the imbalance of an edge e = uV ∈ E as |dG(u)−dG(V)| and the irregularity of G as irr (G) = ∑ dG (u) − dG (V)
Summary
(2) has an expected property of an irregularity measure that graphs with the same degree sequence have the same total irregularity, while (1) does not have Both measures have common properties, including that they are zero if and only if G is regular. In [5], the authors obtained the upper bound of the total irregularity among all graphs with n vertices, and they showed that the star graph Sn is the tree with the maximal total irregularity among all trees with n vertices. (2) If G is a tree, irrt(G) ≤ (n − 1)(n − 2), with equality holds if and only if G ≅ Sn. In [7], the authors investigated the total irregularity of unicyclic graphs and determined the graph with the maximal total irregularity n2 − n − 6 among unicyclic graphs on n vertices.
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