$\vec{\mathcal C}$-Homogeneous Graphs Via Ordered Pencils

A finite connected graph $G_r^\sigma$ is constructed for $3\le r\in\mathbb{Z}$ and $\sigma\in(0,r-1)\cap\mathbb{Z}$ via ordered binary projective pencils, with each copy of $K_2$ as intersection of two induced subgraphs of $G_r^\sigma$ namely: {\bf(i)} a maximal clique $X_0=K_{2s}$ with $s=2^{r-\sigma-1}$ and {\bf(ii)} a maximal Turan graph $X_1=T_{ts,t}$ with $t=2^{\sigma+1}-1$. For $i=0,1$ it holds that $G_r^\sigma$: {\bf(a)} is an edge-disjoint union of copies of $X_i$; {\bf(b)} every copy of $X_i$ in $G_r^\sigma$ shares each of its edges with just one copy of $X_{1-i}$ in $G_r^\sigma$; {\bf(c)} is $K_4$-ultrahomogeneous of order $(2^r-1)(2^r-2)$ with $2^{\sigma+1}$ edge-disjoint copies of $K_4$ at each vertex if $r-\sigma=2$; {\bf(d)} satisfies a weak definition of $\{K_{2s},T_{ts,t}\}$-{\it homogeneous graphs} that include the $\{K_4,T_{6,3}=K_{2,2,2}\}$-ultrahomogeneous graph $C_3^1$, weak in that each induced copy of $X_i$ is preserved with a distinguished arc. Computations for $r\le 8$ and $r-\sigma\le 5$ show $G_r^\sigma$ to be the connected Menger graph of a corresponding configuration.