On the Spectral Radius of Minimally 2-(Edge)-Connected Graphs with Given Size

A graph is minimally $k$-connected ($k$-edge-connected) if it and deleting any arbitrary chosen edge always leaves a which not ($k$-edge-connected). Let $m= \binom{d}{2}+t$, $1\leq t\leq d$ $G_m$ be the obtained from complete $K_d$ by adding one new vertex of degree $t$. $H_m$ $K_d\backslash\{e\}$ adjacent to precisely two vertices $d-1$ in $K_d\backslash\{e\}$. Rowlinson [Linear Algebra Appl., 110 (1988) 43--53.] showed that attains maximum spectral radius among all graphs size $m$. This classic result indicates $2$-(edge)-connected $m=\binom{d}{2}+t$ except $t=1$. The next year, [Europ. J. Combin., 10 (1989) 489--497] proved $2$-connected $m=\binom{d}{2}+1$ ($d\geq 5$), this also unique extremal 5$). Observe neither nor are graphs. In paper, we determine for ($2$-edge-connected) given size; moreover, corresponding characterized.