New results on $B_{\alpha }$-eigenvalues of a graph

Let $G$ be a graph with adjacency matrix $A(G)$ and Laplacian matrix $L(G)$. In 2024, Samanta $\mathit{\text{et}}$ $\mathit{\text{al.}}$ defined the convex linear combination of $A(G)$ and $L(G)$ as $B_{\alpha }(G)=\alpha A(G)+(1-\alpha )L(G)$, for $\alpha \in [0,1]$. This paper presents some results on the eigenvalues of $B_{\alpha }(G)$ and their multiplicity when some sets of vertices satisfy certain conditions. Moreover, the positive semidefiniteness problem of $B_{\alpha }(G)$ is studied.