A c-cyclic graph is a connected graph with n vertices and n + c−1 edges. In this paper, we consider the problem that: in the class of graphs, each of which is the complement graph of a c-cyclic graph with n vertices, which graph has the largest $A_{\alpha }$-spectral radius. We shall show that, for $0\le c\le n-4$, the extremal graph must be a graph with an isolated vertex. This implies that the maximum degree of G is larger, then the $A_{\alpha }$-spectral radius of its complement graph is also larger. We also determine the structure of the extremal graph with largest $A_{\alpha }$-spectral radius among the class of graphs, each of which is the complement graph of a c-cyclic graph with n vertices and maximum degree n−1 for $0\le c\le n-3$ and n−2 for $0\le c\le n-4$, respectively.

c循环图是具有n 个顶点和n  +  c -1 条边的连通图。在本文中，我们考虑的问题是：在一类图中，每一个图都是一个有n个顶点的c-循环图的补图，哪个图最大$A_{\alpha }$-光谱半径。我们将证明，对于$0\le C\le n-\mathrm{4个}$，极值图必须是具有孤立顶点的图。这意味着G的最大次数越大，则$A_{\alpha }$-其补图的谱半径也较大。我们还确定了具有最大的极值图的结构$A_{\alpha }$-图类间的谱半径，每一个图都是n个顶点最大度为n −1的c-循环图的补图$0\le C\le n-\mathrm{3个}$和n -2 对于$0\le C\le n-\mathrm{4个}$， 分别。