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In this article, a series of new inequalities involving the $q$-numerical radius for $n\times n$ tridiagonal, and anti-tridiagonal operator matrices has been established. These inequalities serve to establish both lower and upper bounds for the $q$-numerical radius of operator matrices. Additionally, we developed $q$-numerical radius inequalities for $n\times n$ circulant, skew circulant, imaginary circulant, imaginary skew circulant operator matrices. Important examples have been used to illustrate the developed inequalities. In this regard, analytical expressions and a numerical algorithm have also been employed to obtain the $q$-numerical radii. We also provide a concluding section, which may lead to several new problems in this area., Comment: 23 pages, 9 figures

The main goal of this article is to establish several new $\mathbb{A}$-numerical radius equalities and inequalities for $n\times n$ cross-diagonal, left circulant, skew left circulant operator matrices, where $\mathbb{A}$ is the $n\times n$ diagonal operator matrix whose diagonal entries are positive bounded operator $A$. Also, we introduce two new matrices called left imaginary circulant operator matrix and left imaginary skew circulant operator matrix and present their $\mathbb{A}$-numerical radii. Certain $\mathbb{A}$-numerical radii of general $n\times n$ operator matrices are obtained. Some special cases of our results lead to the results of earlier works in the literature, which shows that our results are more general. Applications of our results are established through some interesting examples. We also provide a concluding section by posing a problem for future research., Comment: 35 pages