On the uniform structure of bipartite graphs admitting a dual adjacency matrix candidate

Let Γ Γ denote a finite, bipartite, connected graph with vertex set X . Fix x ∈ X x ∈ X and let ε ≥ 3 ε ≥ 3 denote the eccentricity of x . For mutually distinct scalars \θ ^*ᵢ\ᵢ₌₀^ε { θ i ∗ } i = 0 ε define a diagonal matrix A^*=A^*(θ ^*₀, θ ^*₁, … , θ ^*ε) ∈ \,Mat\,X(R) A ∗ = A ∗ ( θ 0 ∗ , θ 1 ∗ , … , θ ε ∗ ) ∈ Mat X ( R ) as follows: for y ∈ X y ∈ X set (A^*)yy = θ ^*∂ (x,y) ( A ∗ ) yy = θ ∂ ( x , y ) ∗ , where ∂ ∂ denotes the shortest path-length distance function of Γ Γ . We say that A^* A ∗ is a dual adjacency matrix candidate of Γ Γ