The Edge-Szeged Index and the PI Index of Benzenoid Systems in Linear Time

The edge-Szeged index of a graph $G$ is defined as $Sz_e(G) = \sum_{e=uv \in E(G)}m_u(e)m_v(e)$, where $m_u(e)$ denotes the number of edges of $G$ whose distance to $u$ is smaller than the distance to $v$ and $m_v(e)$ denotes the number of edges of $G$ whose distance to $v$ is smaller than the distance to $u$. Similarly, the PI index is defined as $PI(G) = \sum_{e=uv \in E(G)}(m_u(e) + m_v(e))$. In this paper it is shown how the problem of calculating the indices of a benzenoid system can be reduced to the problem of calculating weighted indices of three different weighted quotient trees. Furthermore, using these results, algorithms are established that, for a given benzenoid system $G$ with $m$ edges, compute the edge-Szeged index and the PI index of $G$ in $O(m)$ time. Moreover, it is shown that the results can also be applied to weighted benzenoid systems.