Irregularity of expansions and Pell graphs

For a graph $G$ the imbalance of an edge $uv$ of $G$ is $\mathrm{\mid }de{g}_G(u)-de{g}_G(v)\mathrm{\mid }$. Irregularity of a graph $G$ is defined as the sum of imbalances over all edges of $G$. In this paper we consider expansions and Pell graphs. If $H$ is an expansion of $G$ with respect to the sets $V_1$ and $V_2$, we express the irregularity of $H$ using $G,V_1$ and $V_2$. For Pell graphs the imbalance of their edges is studied. The number of edges of a Pell graph with a fixed imbalance $k$ is expressed. Using these results the irregularity and the $\sigma$-index of Pell graphs are given.