The total irregularity of a graph

In this note a new measure of irregularity of a simple undirected graph $G$ is introduced. It is named the total irregularity of a graph and is defined as $\irr_t(G) = 1/2\sum_{u,v \in V(G)} |d_G(u)-d_G(v)|$, where $d_G(u)$ denotes the degree of a vertex $u \in V(G)$. The graphs with maximal total irregularity are determined. It is also shown that among all trees of same order the star graph has the maximal total irregularity.

💡 Research Summary

The paper introduces a novel quantitative measure of graph irregularity called the total irregularity, denoted irrₜ(G). For a simple undirected graph G with vertex set V(G) and degree function d_G, the total irregularity is defined as

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