The total irregularity of a graph

In this note a new measure of irregularity of a graph G is introduced. It is named the total irregularity of a graph and is defined as irrt(G) = 1 / 2∑u,v ∈V(G) |dG(u)-dG(v)|, where dG(u) denotes the degree of a vertex u ∈V(G). All graphs with maximal total irregularity are determined. It is also shown that among all trees of the same order the star has the maximal total irregularity

The Maximal Total Irregularity of Bicyclic Graphs

In 2012, Abdo and Dimitrov defined the total irregularity of a graph G=(V,E) as irrtG=1/2∑u,v∈VdGu-dGv, where dGu denotes the vertex degree of a vertex u∈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 on n vertices

On ve-Degree Irregularity Indices

In this paper, vertex-edge degrees (or simply, ve-degrees) of vertices in a graph are considered. The ve-degree of a vertex v in a graph equals to the number of different edges which are incident to a vertex from the closed neighborhood of v. The author introduces the ve-degree total irregularity index here and calculates this index for paths and double star graphs. Finally, the maximal trees are characterized with respect to the ve-degree total irregularity index

Non-regular graphs with minimal total irregularity

The {\it total irregularity} of a simple undirected graph $G$ is defined as ${\rm irr}_t(G) =$ $\frac{1}{2}\sum_{u,v \in V(G)}$ $\left| d_G(u)-d_G(v) \right|$, where $d_G(u)$ denotes the degree of a vertex $u \in V(G)$. Obviously, ${\rm irr}_t(G)=0$ if and only if $G$ is regular. Here, we characterize the non-regular graphs with minimal total irregularity and thereby resolve the recent conjecture by Zhu, You and Yang~\cite{zyy-mtig-2014} about the lower bound on the minimal total irregularity of non-regular connected graphs. We show that the conjectured lower bound of $2n-4$ is attained only if non-regular connected graphs of even order are considered, while the sharp lower bound of $n-1$ is attained by graphs of odd order. We also characterize the non-regular graphs with the second and the third smallest total irregularity

The Minimal Total Irregularity of Graphs

In \cite{2012a}, Abdo and Dimitov defined the total irregularity of a graph $G=(V,E)$ as \hskip3.3cm $\rm irr_{t}$$(G) = \frac{1}{2}\sum_{u,v\in V}|d_{G}(u)-d_{G}(v)|,$ \noindent where $d_{G}(u)$ denotes the vertex degree of a vertex $u\in V$. In this paper, we investigate the minimal total irregularity of the connected graphs, determine the minimal, the second minimal, the third minimal total irregularity of trees, unicyclic graphs, bicyclic graphs on $n$ vertices, and propose an open problem for further research.Comment: 13 pages, 4 figure