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NONREGULAR GRAPHS WITH MINIMAL TOTAL IRREGULARITY

Part of: Graph theory

Published online by Cambridge University Press:  29 April 2015

HOSAM ABDO  and
DARKO DIMITROV
HOSAM ABDO
Affiliation:
Department of Mathematics and Computer Science, Freie Universität Berlin, Takustraße 9, D-14195 Berlin, Germany email abdo@mi.fu-berlin.de
DARKO DIMITROV*
Affiliation:
Department of Engineering Sciences I, Hochschule für Technik und Wirtschaft Berlin, Wilhelminenhofstraße 75A, D-12459 Berlin, Germany email darko.dimitrov11@gmail.com
*
✉darko.dimitrov11@gmail.com
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Abstract

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The total irregularity of a simple undirected graph $G$ is defined as $\text{irr}_{t}(G)=\frac{1}{2}\sum _{u,v\in V(G)}|d_{G}(u)-d_{G}(v)|$, where $d_{G}(u)$ denotes the degree of a vertex $u\in V(G)$. Obviously, $\text{irr}_{t}(G)=0$ if and only if $G$ is regular. Here, we characterise the nonregular graphs with minimal total irregularity and thereby resolve the recent conjecture by Zhu et al. [‘The minimal total irregularity of graphs’, Preprint, 2014, arXiv:1404.0931v1 ] about the lower bound on the minimal total irregularity of nonregular connected graphs. We show that the conjectured lower bound of $2n-4$ is attained only if nonregular connected graphs of even order are considered, while the sharp lower bound of $n-1$ is attained by graphs of odd order. We also characterise the nonregular graphs with the second and the third smallest total irregularity.

Keywords

total irregularityextremal graphs

MSC classification

Primary: 05C07: Vertex degrees
Secondary: 05C35: Extremal problems
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 92 , Issue 1 , August 2015 , pp. 1 - 10
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

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