The edge-deletion energy conjecture for degree-based weighted adjacency matrices

Let $K_n$ be the complete graph, let $e$ be an edge of $K_n$, and let $\boldsymbol{\dot{A}}(K_n)$ and $\boldsymbol{\dot{A}}(K_n-e)$ denote the degree-based weighted adjacency matrices whose edge weights are determined by a function $\phi(x,y)$ of the endpoint degrees. Their energies are the sums of the absolute values of their adjacency-matrix eigenvalues. Edge-deletion energy conjecture. The energy of $\dot{A}(K_n)$ is strictly greater than the energy of $\dot{A}(K_n-e)$ for almost all functions $\phi(x,y)$. This observation is based on computational experiments and numerical comparisons for several well-known degree-based weighted adjacency matrices; the phrase “almost all” is not further quantified in the source, so the precise scope of the claim remains open.