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

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