The clique number of the exact distance $t$-power graph: complexity and eigenvalue bounds

The exact distance $t$-power of a graph $G$, $G^{[\sharp t]}$, is a graph which has the same vertex set as $G$, with two vertices adjacent in $G^{[\sharp t]}$ if and only if they are at distance exactly $t$ in the original graph $G$. We study the clique number of this graph, also known as the $t$-equidistant number. We show that it is NP-hard to determine the $t$-equidistant number of a graph, and that in fact, it is NP-hard to approximate it within a constant factor. We also investigate how the $t$-equidistant number relates to another distance-based graph parameter; the $t$-independence number. In particular, we show how large the gap between both parameters can be. The hardness results motivate deriving eigenvalue bounds, which compare well against a known general bound. In addition, the tightness of the proposed eigenvalue bounds is studied.