$G(\mathbf{g})$ is the Laplacian matrix, where $\mathbf{g}$ is the vector of elements of matrix $G(\mathbf{g})$. The Laplacian matrix $G(\mathbf{g})$ is defined as
$$
G(\mathbf{g})=\sum_{l}g_la_la_l^T.
$$
Where, $g_l$ is an $l^{th}$ element of vector $\mathbf{g}$ and $a_l$ is the $l^{th}$ column of adjacency matrix.
The resistance distance is defined as
$$
\begin{split}
R_{ij} &= (e_{i}-e_{j})^TG(\mathbf{g})^\dagger (e_{i}-e_{j})\\
& = (e_{i}-e_{j})^T\left(G(\mathbf{g})+\frac{1}{n}\mathbf{1}\mathbf{1}^T\right)^{-1} (e_{i}-e_{j})
\end{split}
$$
where $e_i$ denotes the $i^{th}$ unit vector, with $1$ in the $i^{th}$ position and $0$ elsewhere and $\mathbf{1}$ is a vector of ones.
In the paper page 44, section 2.7, it is argued that the function $R_{ij}$ is a convex function with respect to $\mathbf{g}$. A clear proof is not given. How can we prove properly that the function $R_{ij}$ is convex with respect to $\mathbf{g}$?
2026-04-07 15:33:09.1775575989