Volume 36 • Number 3 • 2013   • On Maximally Irregular Graphs Simon Mukwembi Abstract. Let $G$ be a connected graph with maximum degree $\Delta(G)$. The {\it irregularity index} $t(G)$ of $G$ is defined as the number of distinct terms in the degree sequence of $G$. We say that $G$ is {\it maximally irregular} if $t(G)=\Delta(G)$. The purpose of this note, apart from pointing out that every highly irregular graph is maximally irregular, is to establish upper bounds on the size of maximally irregular graphs and maximally irregular triangle-free graphs. 2010 Mathematics Subject Classification: 97K30 Full text: PDF