On the eccentric distance sum of unicyclic graphs with a given matching number

Let $G = (V_G,E_G)$ be a simple connected graph. The eccentric distance sum of $G$ is defined as $ξ^d(G)=\sum_{v \in V_G}\,\varepsilon_G(v)D_G(v),$ where $\varepsilon_G(v)$ is the eccentricity of the vertex $v$ and $D_G(v)=\sum_{u \in V_G}\,d(u,v)$ is the sum of all distances from the vertex $v$. In this paper, we characterize $n$-vertex unicyclic graphs with given matching number having the minimal and second minimal eccentric distance sums, respectively.

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