  Let $G$ be a simple graph with associated diagonal matrix of vertex degrees
$D(G)$, adjacency matrix $A(G)$, Laplacian matrix $L(G)$ and signless Laplacian
matrix $Q(G)$. Recently, Nikiforov proposed the family of matrices
$A_\alpha(G)$ defined for any real $\alpha\in [0,1]$ as
$A_\alpha(G):=\alpha\,D(G)+(1-\alpha)\,A(G)$, and also mentioned that the
matrices $A_\alpha(G)$ can underpin a unified theory of $A(G)$ and $Q(G)$.
Inspired from the above definition, we introduce the $B_\alpha$-matrix of $G$,
$B_\alpha(G):=\alpha A(G)+(1-\alpha)L(G)$ for $\alpha\in [0,1]$. Note that $
L(G)=B_0(G), D(G)=2B_{\frac{1}{2}}(G), Q(G)=3B_{\frac{2}{3}}(G), A(G)=B_1(G)$.
In this article, we study several spectral properties of $ B_\alpha $-matrices
to unify the theories of adjacency, Laplacian, and signless Laplacian matrices
of graphs. In particular, we prove that each eigenvalue of $ B_\alpha(G) $ is
continuous on $ \alpha $. Using this, we characterize positive semidefinite $
B_\alpha $-matrices in terms of $\alpha$. As a consequence, we provide an upper
bound of the independence number of $ G $. Besides, we establish some bounds
for the largest and the smallest eigenvalues of $B_\alpha(G)$. As a result, we
obtain a bound for the chromatic number of $G$ and deduce several known
results. In addition, we present a Sachs-type result for the characteristic
polynomial of a $ B_\alpha $-matrix.
