The general spectral radius of a graph $G$, denoted by $Θ(G,α)$, is the maximal eigenvalue of $M_α(G)=A(G)+αD(G)$ $(α\geq 0)$, where $A(G)$ and $D(G)$ are the adjacency matrix and the diagonal matrix of vertex degrees of $G$, respectively. A graph $G$ is called $Θ_α$-maximal in a class of connected simple graphs $\mathcal {G}$ if $Θ(G,α)$ is maximal among all graphs of $\mathcal {G}$. A $t$-cone $c$-cyclic graph is the join of a complete graph $K_t$ and a $c$-cyclic connected simple graph. Let $π=\big(d_1,d_2,\ldots,d_n\big)$ and $π'=\big(d'_1,d'_2,\ldots,d'_n\big)$ be two non-increasing degree sequences of $t$-cone $c$-cyclic graphs with $n$ vertices. We say $π$ is strictly majorized by $π'$, denoted by $π\lhd π'$, if $π\neq π'$, $\sum_{i=1}^n d_i=\sum_{i=1}^n d_i'$, and $\sum_{i=1}^k d_i\leq \sum_{i=1}^k d_i'$ for $k=1,2,\ldots,n-1$. Denote by $Γ(π,t;c)$ the class of $t$-cone $c$-cyclic graphs with $π$ as its degree sequence. In this paper, we determine some properties of $Θ_α$-maximal graphs of $Γ(π,t;c)$ and characterize the unique $Θ_α$-maximal graph of $Γ(π,t;0)$ \big(resp. $Γ(π,t;1)$ and $Γ(π,t;2)$\big). Moreover, we prove that if $π\lhd π'$, $G$ and $G'$ are the $Θ_α$-maximal graphs of $Γ(π,t;c)$ and $Γ(π',t;c)$ respectively, then $Θ(G,α)<Θ(G',α)$ for $c\in \big\{0,1\big\}$, and we also consider the similar result for $c=2$.