on the $a_{alpha}$-spectrum of the $k$-splitting signed graph and neighbourhood coronas

Let $sigma=(g,sigma)$ be a signed graph with adjacency matrix $a(sigma)$ and $d(g)$ be the diagonal matrix of its vertex degrees. for any real $alphain [0,1]$, the $a_{alpha}$-matrix of a signed graph $sigma$ is defined as $a_{alpha}(sigma)=alpha d(g)+(1-alpha)a(sigma)$. given a signed graph $sigma$ with vertex set $v={v_1, v_2,dots, v_n}$, the $k$-splitting signed graph $sp_k(sigma)$ of $sigma$ is obtained by adding to each vertex $vin v(sigma)$ new $k$ vertices say $u^1, u^2, ldots, u^k$ and joining every neighbour say $u$ of the vertex $v$ to $u^i$, $1le ile k$ by an edge which inherits the sign from $uv$. in this paper, we determine the $a_{alpha}$-spectrum of $sp_k(sigma)$ in case of $sigma$ being a regular signed graph. for $k=1$, we introduce two distinct coronas of signed graphs $sigma_1$ and $sigma_2$ based on $sp_1(sigma_1)$, namely the splitting v-vertex neighbourhood corona and the splitting s-vertex neighbourhood corona. by examining the $a_{alpha}$-characteristic polynomial of the resulting signed graphs, we derive their $a_{alpha}$-spectra under certain regularity conditions on the constituent signed graphs. as applications, we use these results to construct infinite pairs of nonregular $a_{alpha}$-cospectral signed graphs.