Abstract A graph is said to be edge‐superconnected if each minimum edge‐cut consists of all the edges incident with some vertex of minimum degree. A graph G is said to be a $\{d,d+1\}$ ‐semiregular graph if all its vertices have degree either d or $d+1$ . A smallest $\{d,d+1\}$ ‐semiregular graph G with girth g is said to be a $(\{d,d+1\};g)$ ‐cage. We show that every $(\{d,d+1\};g)$ ‐cage with odd girth g is edge‐superconnected. © 2011 Wiley Periodicals, Inc. NETWORKS, 2011

Abstract An ({r, r + 1}; g)‐cage is a graph with degree set {r, r + 1}, girth g, and with the smallest possible order; every such graph is called a semiregular cage. In this article, semiregular cages are shown to be maximally edge‐connected and 2‐connected. As a consequence, ({3, 4}; g)‐cages are proved to be maximally connected. © 2009 Wiley Periodicals, Inc. NETWORKS, 2010