Let $G$ be a graph with adjacency matrix $A(G)$ and Laplacian matrix $L(G)$. In 2024, Samanta \textit{et} \textit{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.

翻译：设$G$是一个图，其邻接矩阵为$A(G)$，拉普拉斯矩阵为$L(G)$。2024年，Samanta等人定义了$A(G)$与$L(G)$的凸线性组合为$B_\alpha(G) = \alpha A(G) + (1-\alpha)L(G)$，其中$\alpha \in [0,1]$。本文给出了当某些顶点集满足特定条件时，$B_{\alpha}(G)$的特征值及其重数的一些结果。此外，还研究了$B_{\alpha}(G)$的半正定性问题。