The purpose of this paper is twofold. Firstly, we generalize the notion of semigeneric braid arrangement to semigeneric graphical arrangement. Secondly, we give a formula for the characteristic polynomial of the semigeneric graphical arrangement ${\mathcal{S}}_G:=\{x_i-x_j=a_i:ij\in E(G),1\le i<j\le n\}$, where $a_i{}^{\prime }\mathrm{s}$ are generic. The formula is obtained by characterizing central subarrangements of ${\mathcal{S}}_G$ via the corresponding graph $G$.

A central arrangement $\mathcal{A}$ was termed free if the module of $\mathcal{A}$-derivations was a free module. The combinatorial structure of arrangements was heavily influenced by the freeness. Yet, there has been scarce exploration into the construction of their bases. In this paper, we constructed the explicit bases for a class of free arrangements that positioned between the cone of Linial arrangements and Shi arrangements.