On the number of solutions of systems of diagonal equations through diagonal GP-graphs: the general and the Hermitian-form cases

For any $m,s\in \mathbb{N}$, we study the number $N_{m\times s,q}(\kappa ,\beta )$ of solutions $(x_1,\dots ,x_s)\in ({\mathbb{F}}_q{)}^s$ of the monic system of diagonal equations $$X_1^{k_i}+\cdots +X_s^{k_i}={\beta }_i,(1\le i\le m),$$ with $\kappa =(k_1,\dots ,k_m)\in {\mathbb{N}}^m$ and $\beta =({\beta }_1,\dots ,{\beta }_m)\in ({\mathbb{F}}_q{)}^m$. We show that this number can be obtained in terms of some data of $\mathit{\text{diagonal}}$ GP-graphs $\mathrm{\Gamma }(\kappa ,q)$. This is a new family of graphs that we introduce here, i.e. Cayley graphs of the form $$\mathrm{\Gamma }(\kappa ,q)=Cay({\mathbb{F}}_q^m,R_{\kappa })\text{where}{R}_{\kappa }=\{(x^{k_1},\dots ,x^{k_m}):x\in {\mathbb{F}}_q^{\ast }\},$$ with $\kappa =(k_1,\dots ,k_m)\in {\mathbb{N}}^m$. In particular, we give three different expressions for $N_{m\times s,q}(\kappa ,\beta )$: one in terms of walks, another in terms of adjacency matrices of $\mathrm{\Gamma }(\kappa ,q)$ and the last one in terms of the spectrum of $\mathrm{\Gamma }(\kappa ,q)$. Finally, we explicitly derive combinatorial formulas for the number of solutions $N_m(s,q)=N_{m\times s,q}({\kappa }_{\mathrm{\ell }},0)$ of monic homogeneous systems of diagonal equations of the form $$X_1^{q^{{\mathrm{\ell }}_i}+1}+\cdots +X_s^{q^{{\mathrm{\ell }}_i}+1}=0(1\le i\le m),$$ with ${\kappa }_{\mathrm{\ell }}=({\mathrm{\ell }}_1,\dots ,{\mathrm{\ell }}_m)=(1,3,\dots ,2m-1)$ and $m\ge 2$, via the known spectrum of Hermitian-form graphs, which can be viewed as diagonal GP-graphs. For any $m,s\in \mathbb{N}$, we give general summation and recursive formulas for $N_m(s,q)\in \mathbb{Z}[q]$. For the small cases $N_1(s,q)$, $N_2(s,q)$ and $N_m(s,q)$, with $1\le s\le 5$, we give explicit expressions.