When $b = 0$, we have an $n \times n$ symmetric arrowhead matrix. When $b \neq 0$, we have

$$\begin{bmatrix} c & c & c & \cdots & c & c \\ c & a & b & \cdots & b & b \\ c & b & a & \cdots & b & b \\ c & b & b & \cdots & b & b \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ c & b & b & \cdots & a & b \\ c & b & b & \cdots & b & a \\ \end{bmatrix} = \begin{bmatrix} c-b & c-b & c-b & \cdots & c-b & c-b \\ c-b & a-b & 0 & \cdots & 0 & 0 \\ c-b & 0 & a-b & \cdots & 0 & 0 \\ c-b & 0 & 0 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ c-b & 0 & 0 & \cdots & a-b & 0 \\ c-b & 0 & 0 & \cdots & 0 & a-b \\ \end{bmatrix} + b \, 1_n 1_n^{\top}$$

which is the sum of a symmetric arrowhead matrix and a (nonzero) constant matrix.

When $b = 0$, we have an $n \times n$ symmetric arrowhead matrix. When $b \neq 0$, we have

$$\begin{bmatrix} c & c & c & \cdots & c & c \\ c & a & b & \cdots & b & b \\ c & b & a & \cdots & b & b \\ c & b & b & \cdots & b & b \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ c & b & b & \cdots & a & b \\ c & b & b & \cdots & b & a \\ \end{bmatrix} = \begin{bmatrix} c-b & c-b & c-b & \cdots & c-b & c-b \\ c-b & a-b & 0 & \cdots & 0 & 0 \\ c-b & 0 & a-b & \cdots & 0 & 0 \\ c-b & 0 & 0 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ c-b & 0 & 0 & \cdots & a-b & 0 \\ c-b & 0 & 0 & \cdots & 0 & a-b \\ \end{bmatrix} + b \, 1_n 1_n^{\top}$$

which is the sum of a symmetric arrowhead matrix and a (nonzero) multiple of the all-ones matrix.