\(\lambda’\)-Optimal Regular Graphs with Two Orbits

Abstract

An edge set \(F\)is called a restricted edge-cut if \(G – F\) is disconnected and contains no isolated vertices. The minimum cardinality over all restricted edge-cuts is called the restricted edge-connectivity of \(G\), and denoted by \(\lambda'(G)\). A graph \(G\) is called \(\lambda’\)-optimal if \(\lambda'(G) = \xi(G)\), where \(\xi(G) = \min\{d_G(u) + d_G(v) – 2: uv \in E(G)\}\). In this note, we obtain a sufficient condition for a \(k( \geq 3)\)-regular connected graph with two orbits to be \(\lambda’\)-optimal.