--- title: 'Queen Graphs: Movement, Domination, and Spectral Theory' url: https://www.emergentmind.com/topics/queen-graphs type: topic --- # Queen Graphs: Movement, Domination, and Spectral Theory Queen graphs are graph-theoretic models of queen movement on chessboards and related lattice boards. In the basic two-dimensional case, the vertex set is the set of board squares, and two vertices are adjacent exactly when the corresponding squares lie in a common row, column, or diagonal, so that a queen can move between them in one move. This family extends naturally to rectangular boards, toroidal boards, higher-dimensional boards, and constrained lattice regions such as polyominoes; across these settings, queen graphs have been studied through domination, independence, edge-coloring, chip-firing, spectral analysis, and exact computation [1606.02060][2604.03842][1810.10961][2312.04686]. ## 1. Definitions, variants, and geometric models The ordinary queen graph on an \(n\times n\) board is the graph whose vertices are the board squares and whose edges encode legal queen moves. In coordinate form, adjacency between \((i,j)\) and \((i',j')\) is given by one of \[ i=i',\qquad j=j',\qquad i-j=i'-j',\qquad i+j=i'+j'. \] This same definition appears in the rectangular queen’s graph \(Q_{m\times n}\), where the board need not be square, and in the notation \(Q_{m,n}\) used in edge-coloring work; in both cases the graph is the union of a rook graph and a bishop graph, with row/column edges supplied by rook moves and diagonal edges supplied by bishop moves [1606.02060][1606.07918]. Several nonclassical variants are now standard. In the toroidal two-dimensional setting, the board is \(\mathbb Z_n^2\), diagonal adjacency is interpreted modulo \(n\), and the resulting graph is the toroidal queen graph. In higher dimension, the three-dimensional queen graph \(Q_n^3\) has vertex set \[ [n]^3=\{0,1,\dots,n-1\}^3, \] with adjacency determined by legal 3D queen moves along 13 undirected line families: 3 axis directions, 6 face diagonals, and 4 space diagonals. A toroidal 3D analogue replaces \([n]^3\) by \((\mathbb Z_n)^3\) and wraps those lines around the discrete torus [2604.03793][2604.03842]. A different generalization arises from restricted regions. For a polyomino \(P\), one may define a queen visibility graph \(Q(P)\) whose vertices are the tiles of \(P\), with adjacency whenever two tiles lie on a common row-, column-, or diagonal-line segment fully contained in \(P\). This is not the ordinary full-board queen graph, because holes, bottlenecks, and interrupted line of sight alter adjacency [1810.10961]. A further directed variant appears in Catalan queen-path models, where allowed moves are only east, north, and northeast by arbitrary positive distance, yielding a directed lattice graph rather than an undirected attack graph [1109.1806]. This diversity of definitions is not merely notational. Boundary effects, periodicity, and dimensionality change both local degree structure and global invariants. A plausible implication is that “queen graph” is best understood as a family of chess-move graphs rather than a single graph class. ## 2. Domination, coverage, and related extremal problems Domination is one of the central themes in queen-graph theory. For a queen graph \(G\), a dominating set is a subset \(D\subseteq V(G)\) such that every vertex lies in \(D\) or is adjacent to a vertex of \(D\). On rectangular boards, the domination number \(\gamma(Q_{m\times n})\) has been computed for \[ 4\le m\le n\le 18, \] and the general lower bound \[ \gamma(Q_{m\times n}) \ge \min\left\{m,\left\lceil \frac{m+n-2}{4} \right\rceil\right\} \] was established. The same work showed that monotonicity can fail: for example, \[ \gamma(Q_{8\times 11}) = 6 > 5 = \gamma(Q_{9\times 11}) = \gamma(Q_{10\times 11}) = \gamma(Q_{11\times 11}), \] so enlarging the board can unlock more efficient geometric configurations rather than making domination harder [1606.02060]. The three-dimensional domination problem is substantially different. For the cubic board \(Q_n^3\), the domination number satisfies \[ \gamma(Q_n^3)\ge \left\lceil \frac{n^3}{13n-12}\right\rceil \] from the maximum closed-neighborhood size, and also \[ \gamma(Q_n^3)\le n\cdot \gamma(Q_n^2)\le \frac{69}{133}n^2+O(n), \] yielding \[ \gamma(Q_n^3)=\Theta(n^2). \] The paper “On the Structure of 3D Queen Domination” identifies an inner core \[ C=\{1,\dots,n-2\}^3 \] and studies \[ \kappa(q)=|N[q]\cap C|, \] the number of core vertices dominated by a queen at \(q\). It proves a strict hierarchy by position type: \[ \kappa(q)=m \text{ (corner)},\qquad \kappa(q)=2m-1 \text{ (edge)}, \] \[ \kappa(q)\le 5m-4-\varepsilon_m \text{ (face)},\qquad \kappa(q)\le 13m-12 \text{ (interior)}, \] with \(m=n-2\) and \(\varepsilon_m=(m-1)\bmod 2\). In particular, no boundary queen matches the best inner-core coverage of an interior queen [2604.03793]. Coverage-maximization with a fixed number of queens is distinct from domination minimization. In “Thresholds of Queen covers,” a \(q\)-configuration \(\mathcal Q\) is evaluated by \[ cover_n(\mathcal Q)=[\mathcal Q\cup A(\mathcal Q)]\cap {}_n, \] and for sufficiently large \(n\), \[ cover_n(\mathcal Q)=(4n-3)q-loss(\mathcal Q), \] where \[ loss(\mathcal Q)=inloss(\mathcal Q)+cenloss(\mathcal Q). \] Here \(inloss\) measures redundant attacks and \(cenloss\) measures boundary truncation from off-center placement. The paper proves that for every fixed \(q\), there is a threshold beyond which every optimal \(q\)-configuration is non-attacking, and a second threshold beyond which the set of optimal configurations stabilizes [2508.02545]. This sharply separates maximum-coverage problems from minimum-domination problems: attacking queens may be useful on small boards, but asymptotically they become inefficient. For polyominoes and polycubes, queen guarding is again a domination problem, now in the induced visibility graph. Every \(n\)-tile \(d\)-polycube can be guarded by at most \(\lfloor n/3\rfloor\) queens, and this bound is sometimes necessary; moreover, deciding whether a polyomino can be guarded by at most \(m\) queens is NP-hard [1810.10961]. ## 3. Independence, gonality, and nonattacking configurations Independent sets in queen graphs are exactly nonattacking queen placements. On the ordinary square board, the classical \(n\)-queens problem asks whether \(\alpha(\mathcal Q(n))\ge n\), and the spectral survey of \(\mathcal Q(n)\) recalls the exact values \[ \alpha(\mathcal Q(n))= \begin{cases} 1, & n=2,\\ 2, & n=3,\\ n, & n\ge 4. \end{cases} \] For rectangular boards \(Q_{n,m}\), the gonality paper uses the corresponding independence numbers \[ \alpha(Q_{2,2})=1,\qquad \alpha(Q_{3,3})=2, \] and for all other boards \[ \alpha(Q_{n,m})= \begin{cases} n & \text{if } n=m\ge 4,\\ \min\{n,m\}=n & \text{if } n\ne m. \end{cases} \] Thus, except for the two smallest square exceptions, the maximum size of a nonattacking set is the smaller board dimension [2012.01992][2312.04686]. A particularly strong connection between independence and chip-firing is the exact gonality formula \[ \operatorname{gon}(Q_{n,m})=nm-\alpha(Q_{n,m}), \] with the explicit specialization \[ \operatorname{gon}(Q_{n,m})= \begin{cases} 3 & \text{if } n=m=2,\\ 7 & \text{if } n=m=3,\\ n(m-1) & \text{otherwise.} \end{cases} \] The same identity holds for toroidal queen’s graphs: \[ \operatorname{gon}(TQ_{n,m})=nm-\alpha(TQ_{n,m}). \] Moreover, maximum independent sets \(S\) correspond bijectively to gonality-achieving divisor classes through \[ S\mapsto [\mathbb 1_{S^c}], \] and distinct maximum independent sets give distinct divisor classes [2312.04686]. In this setting, nonattacking queen placements are not merely combinatorial witnesses; they classify all minimal positive-rank divisors. Toroidal independence becomes subtler in higher dimension. On \(\mathbb Z_n^d\), queens attack along every toroidal line whose direction vector lies in \(\{-1,0,1\}^d\setminus\{0\}\). The trivial upper bound is \[ \alpha(Q_{n,d}^{\mathrm{tor}})\le n^{d-1}. \] If \(n\) has no prime divisor less than \(2^d\), then this upper bound is attained: \[ \alpha(Q_{n,d}^{\mathrm{tor}})=n^{d-1}. \] But in dimension \(3\), if \(n\) is coprime to \(2\cdot 3\), divisible by \(5\), and not by \(25\), then \[ \alpha(Q_{n,3}^{\mathrm{tor}})