Seidel Energy of Iterated Line Graphs of Regular Graphs

Harishchandra S. Ramane, Ivan Gutman, Mahadevappa M. Gundloor

The Seidel matrix $S(G)$ of a graph $G$ is the square matrix whose $(i,j)$-entry is equal to $-1$ or $1$ if the $i$-th and $j$-th vertices of $G$ are adjacent or non-adjacent, respectively, and is zero if $i=j$. The Seidel energy of $G$ is the sum of the absolute values of the eigenvalues of $S(G)$. We show that if $G$ is regular of order $n$ and of degree $r\geq3$, then for each $k\geq2$, the Seidel energy of the $k$-th iterated line graph of $G$ depends solely on $n$ and $r$. This result enables the construction of pairs of non-cospectral, Seidel equienergetic graphs of the same order.