Certain Types of Total Irregularities of Graphs and Digraphs

The total irregularity of a simple undirected graph $G$ is denoted by $irr_t(G)$ and is defined as $irr_t(G) = \frac{1}{2}\sum\limits_{u,v \in V(G)}|d(u) - d(v)|$. In this paper, the concept called edge-transformation in relation to total irregularity of simple undirected graphs with at least one cut edge is introduced. We also introduce the concept of an edge-joint between two simple undirected graphs. We also introduce the concept of total irregularity in respect of in-degree and out-degree in simple directed graphs. These invariants are called total in-irregularity and total out-irregularity respectively. In this paper, we initiate a study on these parameters of given simple undirected graphs and simple digraphs.

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