\(c\lambda\)-Optimally Connected Mixed Cayley Graph

Abstract

The cyclic edge-connectivity of a cyclically separable graph \(G\), denoted by \(c\lambda(G)\), is the minimum cardinality of all edge subsets \(F\) such that \(G – F\) is disconnected and at least two of its components contain cycles. Since \(c\lambda(G) \leq \zeta(G)\), where \(\zeta(G) = \min\{w(A) \mid A \text{ induces a shortest cycle in } G\}\), for any cyclically separable graph \(G\), a cyclically separable graph \(G\) is said to be cyclically optimal if \(c\lambda(G) = \zeta(G)\). The mixed Cayley graph is a kind of semi-regular graph. The cyclic edge-connectivity is a widely studied parameter, which can be used to measure the reliability of a network. Because previous work studied cyclically optimal mixed Cayley graphs with girth \(g \geq 5\), this paper focuses on mixed Cayley graphs with girth \(g < 5\) and gives some sufficient and necessary conditions for these graphs to be cyclically optimal.