Example. The weighted four-cycle Laplacian [fdip-004B]

✍️sourceAGENTDRAFTED

Use the alternating four-cycle, with vertices \(0,1,2,3\), edges \(e_i:i\to i+1\), and positive real weights \((1,t,1,t)\). For a field \(f=(a,b,c,d)\), the incidence values are \(Df=(b-a,c-b,d-c,a-d)\). The counting adjoint sends an edge field to \[D^*(u_0,u_1,u_2,u_3) =(u_3-u_0,u_0-u_1,u_1-u_2,u_2-u_3).\] First multiply by \(W_t=\operatorname {diag}(1,t,1,t)\), then apply \(D^*\). The resulting \(L_t=D^*W_tD\) has coordinate matrix \[L_t=\begin {pmatrix} 1+t&-1&0&-t\\ -1&1+t&-t&0\\ 0&-t&1+t&-1\\ -t&0&-1&1+t \end {pmatrix}.\]

For example, its first coordinate is \(t(a-d)-(b-a)=(1+t)a-b-td\); the other coordinates follow from the same return formula. Thus \[L_t\delta _0=(1+t,-1,0,-t),\qquad L_t\mathbf 1=0,\] and the counting energy is \[\langle f,L_tf\rangle =(b-a)^2+t(c-b)^2+(d-c)^2+t(a-d)^2.\] The counting adjoint \(D^*\) does not depend on \(t\). The weighted return map \(D^*W_t\), and hence \(L_t\), do. When \(t=1\) the two return maps coincide; otherwise the factors \(t\) must be retained.

This calculation substitutes the four-cycle data into Definition [fdip-0007]; its incidence convention follows [spielman2009effective, Section 2.1, p. 4]. The cyclic energy factorization gives the corresponding operator identity.