Let \(\mathcal {G}\) be a metric non-compact connected graph with finitely many edges. The main object of the paper is the Hamiltonian \(\mathbf{H}_{\alpha }\) associated in \(L^2(\mathcal {G};\mathbb {C}^m)\) with a matrix Sturm–Liouville expression and boundary delta-type conditions at each vertex. Assuming that the potential matrix is summable and applying the technique of boundary triplets and the corresponding Weyl functions, we show that the singular continuous spectrum of the Hamiltonian \(\mathbf{H}_{\alpha }\) as well as any other self-adjoint realization of the Sturm–Liouville expression is empty. We also indicate conditions on the graph ensuring pure absolute continuity of the positive part of \(\mathbf{H}_{\alpha }\). Under an additional condition on the potential matrix, a Bargmann-type estimate for the number of negative eigenvalues of \(\mathbf{H}_{\alpha }\) is obtained. Additionally, for a star graph \(\mathcal {G}\) a formula is found for the scattering matrix of the pair \(\{\mathbf{H}_{\alpha }, \mathbf{H}_D\}\), where \(\mathbf{H}_D\) is the Dirichlet operator on \(\mathcal {G}\).

令\（\ mathcal {G} \）是具有有限多个边的度量非紧连接图。本文的主要对象是哈密顿量\（\ mathbf {H} _ {\ alpha} \）与\（L ^ 2（\ mathcal {G}; \ mathbb {C} ^ m）\）相关联并具有矩阵每个顶点处的Sturm-Liouville表达式和边界增量类型条件。假设势矩阵是可加的，并应用边界三元组和相应的Weyl函数的技术，我们表明哈密顿量\（\ mathbf {H} _ {\ alpha} \的奇异连续谱以及任何其他自-Sturm-Liouville表达式的伴随实现为空。我们还在图上指出确保纯正部分绝对绝对连续的条件\（\ mathbf {H} _ {\ alpha} \）。在势矩阵的附加条件下，获得\（\ mathbf {H} _ {\ alpha} \）的负特征值数量的Bargmann型估计。另外，对于星形图\（\ mathcal {G} \），找到了对\（\ {\ mathbf {H} _ {\ alpha}，\ mathbf {H} _D \} \的散射矩阵的公式），其中\（\ mathbf {H} _D \）是\（\ mathcal {G} \）上的Dirichlet运算符。