a new measure for transmission irregularity extent of graphs

The transmission of a vertex ${varsigma}$ in a connected graph $mathcal{j}$ is the sum of distances between ${varsigma}$ and all other vertices of $mathcal{j}$. a graph $mathcal{j}$ is called transmission regular if all vertices have the same transmission. in this paper, we propose a new graph invariant for measuring the transmission irregularity extent of transmission irregular graphs. this invariant which we call the total transmission irregularity number (tti number for short) is defined as the sum of the absolute values of the difference of the vertex transmissions over all unordered vertex pairs of a graph. we investigate some lower and upper bounds on the tti number which reveal its connection to a number of already established indices. in addition, we compute the tti number for various families of composite graphs and for some chemical graphs and nanostructures derived from them.