GraphTheory-VL

Introduce \(G\) as the simple graph with labeled vertices \(A, B, C, D, E,\) and \(F\), and let \(M\) be its graphic matroid. The supplied rendering is the complete combinatorial specification of \(G\): each black segment joining two labeled points is an edge, a dashed portion continues an occluded black segment, the translucent gray and cyan surfaces contribute no edges, and a visual crossing without a labeled vertex does not create a new vertex. Read the rendering carefully and recover its full edge topology. Order the edges lexicographically by their unordered endpoint pairs using \(A < B < C < D < E < F\). For a spanning tree \(T\) of \(G\) and an edge \(e\) in \(T\), delete \(e\) from \(T\) to obtain its fundamental cut. Call \(e\) internally active when \(e\) is the lexicographically smallest edge in that cut. If \(i(T)\) denotes the number of internally active edges of \(T\), define the internal-activity polynomial by \(P_G(x) = \sum_T x^{i(T)}\), where the sum is over all spanning trees of the graph recovered from the rendering. Determine \(P_G(x)\) exactly, with integer coefficients and powers written in descending order. Your solution must explain how the hidden dashed continuations affect the edge set, identify the resulting graph structurally, and justify the polynomial through a matroid or graph-theoretic computation. Do not infer extra edges from the geometric placement of segments, and do not discard an edge merely because part of it lies behind one of the displayed surfaces. The answer is one polynomial, not a list of values or a choice among alternatives.

Designate \(G\) and \(H\) as the finite simple graphs encoded by the two drawings, with \(G\) listed first and \(H\) listed second. Every dark circular mark is one vertex, and every straight segment joining two marked vertices is one edge. A crossing without a marked vertex does not create a new vertex, so the vertical segment in the lower drawing passes through the horizontal segment without subdivision. The placement, thickness, and apparent length of a segment have no further meaning, and neither graph has loops or multiple edges.

Read all adjacency relations from the complete drawings. For a connected finite graph \(X\), let \(L_X=D_X-A_X\) be its integer Laplacian, where \(D_X\) is the diagonal degree matrix and \(A_X\) is the adjacency matrix. Delete one row and the corresponding column from \(L_X\), obtaining a reduced Laplacian \(\widetilde L_X\), and define the critical group by \[ K(X)=\mathbb Z^{|V(X)|-1}/\widetilde L_X\mathbb Z^{|V(X)|-1}. \] The isomorphism type does not depend on the deleted vertex. Identify the structural graph forms visible in the upper and lower drawings, then determine the Smith normal form data of both reduced Laplacians. Give the ordered pair \((K(G),K(H))\), using the upper graph first. Each group must be expressed in invariant-factor form as a direct sum of cyclic groups \(\mathbb Z/d\mathbb Z\), with every displayed modulus greater than \(1\) and each modulus dividing the next one. Omit trivial cyclic factors and preserve the order of the two graphs in the final answer.

In this setup, consider \(Q\) to be the weighted directed quiver on the ordered vertex set \(V=\{1,2,3,4,5,6\}\), with the arrow directions and multiplicities encoded by the complete visual data supplied with this question. Every arrow without a printed multiplicity has multiplicity 1, while the visibly printed multiplicity 2 is to be used exactly as shown. An absent directed edge contributes zero. Do not replace \(Q\) by its underlying undirected graph, and do not identify two opposite arrows before applying the signed convention below.

Define the skew-symmetric integer matrix \(B=(b_{ij})\) by taking \(b_{ij}\) to be the multiplicity of arrows from \(i\) to \(j\) minus the multiplicity of arrows from \(j\) to \(i\). Thus, the ordering of rows and columns is exactly \(1,2,3,4,5,6\), and the sign depends on the arrow direction, not on the geometric slope or position of an edge. Work over the rational numbers, so that the nullspace is a vector space over \(\mathbb{Q}\).

The visual data are indispensable because the edge directions, the single non-unit multiplicity, and the vertex ordering jointly determine \(B\). From this matrix, consider the nullspace \(K=\ker(B)\). To eliminate the usual nonuniqueness of a basis, impose the following normalization. The free coordinates are listed in increasing index order; for each free coordinate, set it equal to 1, set every other free coordinate equal to 0, and solve for the pivot coordinates. The resulting vectors are written as six-entry column vectors in the vertex order above.

Determine the ordered pair consisting of the normalized nullspace basis vector associated with the first free coordinate and the normalized nullspace basis vector associated with the second free coordinate. Your response must be an exact ordered pair, not a numerical approximation, and must preserve the signs and coordinate order. No additional invariant, rank, determinant, or explanatory text is part of the requested answer.

Throughout, use \(G\) as the embedded planar graph formed by the thick black orthogonal mesh. Use the complete supplied drawing as the data for reconstructing \(G\), but impose the following precise conventions. Only the black mesh belongs to \(G\). The orange, blue, and magenta rings are auxiliary curves and are not graph edges, vertices, or additional boundaries. The thin large circular outline surrounding the construction is also not an edge of \(G\). Whenever two black tracks meet, their meeting point is one vertex, including vertices on the outer boundary of the black mesh. Portions of a black track hidden beneath a colored ring are understood to continue straight through; the colored overdraw does not split the track or create a new vertex. Tangencies or crossings involving only colored curves are ignored.

The black skeleton is a topological rectangular lattice drawn obliquely: it has two transverse track directions and a finite polygonal boundary. Let \(r\) be the number of elementary black segments along one boundary chain, and let \(s\) be the corresponding number along the transverse boundary chain. These counts must be obtained by tracing the black mesh across the entire drawing, including its upper and lower portions; do not count a cropped local patch or use the colored rings as a substitute for black lattice edges. Let \(V\) be the total number of black-mesh vertices, let \(E\) be the total number of black-mesh edges after the hidden portions are restored, and let \(F\) be the number of connected components of the plane minus \(G\), including the unbounded component. Determine the ordered integer triple \((V,E,F)\).