--- title: Signless Laplacian Spectral Radius url: https://www.emergentmind.com/topics/signless-laplacian-spectral-radius type: topic --- # Signless Laplacian Spectral Radius The signless Laplacian spectral radius is a central spectral invariant in algebraic and extremal graph theory, as well as in the study of hypergraphs and simplicial complexes. For a simple graph \( G = (V, E) \), the signless Laplacian matrix is defined as \( Q(G) = D(G) + A(G) \), where \( D(G) \) is the diagonal matrix of vertex degrees and \( A(G) \) is the adjacency matrix. The spectral radius of \( Q(G) \), denoted \( q(G) \), is its largest eigenvalue, guaranteed by Perron–Frobenius theory to correspond to a unique positive eigenvector for connected graphs. In the hypergraph setting, the signless Laplacian becomes an order-\( k \) symmetric tensor, and its spectral radius, again denoted \( q(G) \), is defined as the maximum modulus among eigenvalues with respect to a variational tensor-eigenpair framework. The signless Laplacian spectral radius governs extremal questions on cliques, forbidden subgraphs, matchings, factor structure, spanning trees, minors, and topological properties across graph, hypergraph, and simplicial complex domains. ## 1. Formal Definitions and Tensor Generalizations For a graph \( G=(V,E) \), with \( |V|=n \): - \( Q(G) = D(G) + A(G) \), \( D(G) \) the diagonal of degrees, \( A(G) \) the adjacency matrix. - The signless Laplacian spectral radius \( q(G) \) is the largest eigenvalue of \( Q(G) \); equivalently, \( q(G) = \max_{x\neq 0} \frac{x^\top Q(G)x}{x^\top x} \), and for connected \( G \) there is a unique positive unit eigenvector (Perron vector). For \( k \)-uniform hypergraphs \( G = (V, E) \), the signless Laplacian is a symmetric order-\( k \) tensor \( Q(G) = D(G) + A(G) \), with: - \( A(G) \) encoding edge incidence (normalized as \( 1/(k-1)! \) if \( \{i_1,...,i_k\} \in E \)), \( D(G) \) diagonal in degree. - Tensor eigenpair \( (\lambda, x) \) satisfies \( Q(G) x^{k-1} = \lambda x^{[k-1]} \), with \( x^{[k-1]}_i = x_i^{k-1} \). - The spectral radius \( q(G) \) is the maximum modulus among eigenvalues and is attained for a nonnegative principal eigenvector by nonnegative tensor theory [1807.05955], [2007.13282], [2601.08595]. ## 2. Key Structural Extremal Results and Constructions ### Graphs - For forbidden subgraphs and trees: For graphs omitting any tree on \( 2k+2 \) vertices, the extremal structure is \( S_{n,k} = K_k \vee \overline{K}_{n-k} \), and for trees of size \( 2k+3 \), the extremal graph is \( S_{n,k}^{+} \) (i.e., a join with one extra edge among the independent set). Equality in spectral radius is characterized uniquely by these constructions for large enough \( n \), with the closed formula \( q(S_{n,k}) = \frac{n + 2k - 2 + \sqrt{(n + 2k - 2)^2 - 8(k^2 - k)}}{2} \) [2209.03120]. - For \( K_{s,t} \)-minor free graphs: The extremal graphs are \( F_{s,t}(n) = K_{s-1} \vee (p \cdot K_t \cup K_r) \) with sharp upper bound for signless Laplacian spectral radius and explicit characterization, particularly for \( t=3 \) [1908.04221]. - For forbidden cycles or fans (\( F_k \)), the unique maximizer is the complete split graph \( S_{n,k} \) [2009.04738], and for forbidden intersecting odd cycles the extremal configuration is also \( K_t \vee \overline{K}_{n-t} \) with precise spectral bounds [2108.03895], [2412.08417]. - For planar graphs, the join construction \( K_2 \nabla P_{n-2} \) achieves the maximal signless Laplacian spectral radius among all planar graphs of large order [1407.5170]. ### Hypergraphs - For \( k \)-uniform supertrees, among all with prescribed edge count \( m \) and diameter \( d \), the unique maximizer for signless Laplacian spectral radius is constructed by attaching all excess edges "in the middle" (at \( v_{\lfloor d/2 \rfloor + 1} \)); for constraints on number of pendent edges or pendent vertices, the extremal structure is characterized as a star with certain paths attached and maximal degree sequences realized by BFS supertrees [1807.05955]. - Spectral Turán-type problems for hypergraphs: The signless Laplacian spectral Turán extremal problem can be reduced, under degree-stability, to multipartite extremal families, e.g., the unique balanced complete bipartite \( 3 \)-graph for Fano plane-free hypergraphs yields sharp bounds on signless Laplacian spectral radius [2601.08595]. - For general hypergraphs, the lower bound \( q(G) \ge 2 \sum_{s \in R} \binom{\omega - 1}{s - 1} \), where \( \omega \) is the clique number and \( R \) the set of allowed edge sizes, is sharp (with equality for the complete \( R \)-hypergraph of size \( \omega \)) [2007.13282]. ### Simplicial Complexes - For a pure \( r \)-complex free of top \( r \)-dimensional holes, the maximal signless Laplacian spectral radius among all such complexes is achieved uniquely by the "tented" complex (all facets containing a fixed vertex), with bound \( r n - r^2 + 1 \) [2507.22518]. ## 3. Spectral Bounds, Inequalities, and Characterizations - For a graph \( G \) with vertex degrees \( d_i \), classical and refined bounds include: - \( q(G) \le \lambda_1(G) + \Delta \) (adjacency spectral radius plus maximal degree) [1308.3427]. - Merris-Feng-Yu bound: \( q(G) \le \max_{v} \{ d(v) + \frac{1}{d(v)} \sum_{w \in N(v)} d(w) \} \), with equality only for regular graphs [1308.3427], [1612.03538]. - Bounds in terms of clique number: for connected \( G \) of order \( n \), clique number \( \omega \), the extremal Turán graph achieves the upper bound [1209.3214]. - For strongly connected digraphs, sharp bounds exist in terms of out-degree sequence, clique number, girth, and connectivity, with explicit formulas and cases of equality for extremal constructions (directed cycles, complete digraphs, attached paths) [1401.7091]. - For distance signless Laplacian, transmission-regularity characterizes the equality cases [1308.3427]. ## 4. Extremal Problems and Turán-Type Phenomena - Turán-type extremal problems (maximizing edges or spectral radius under forbidden subgraphs) admit spectral versions: - Under sufficiently large signless Laplacian spectral radius, one guarantees the existence of many cliques or large blowups (spectral supersaturation), stability theorems aligning with the Turán graph as extremal [2507.02263]. - Analogous results for hypergraphs and simplicial complexes: spectral bounds translate into face-number (count of \( r \)-faces) bounds, e.g., for hole-free pure \( r \)-dimensional complexes, Turán-type upper bounds on \( f_r(K) \) follow from spectral extremality [2507.22518], [2601.08595]. ## 5. Applications to Spanning Trees, Matchings, and Factors - Thresholds for existence of perfect matchings or spanning trees with bounded leaf-degree are characterized by explicit spectral bounds: - For perfect matching existence, the threshold given by the spectral radius of specific join constructions is sharp; similar explicit cubic (or quadratic) thresholds apply for every case [2007.04479], [2303.16687], [2406.07132]. - For \( k \)-extendability, cubic and quadratic spectral bounds determine sharp existence conditions; extremal non-\( k \)-extendable graphs attain equality in the bound [2303.16687]. - For spanning trees with leaf degree at most \( k \), spectral threshold \( q(G) > \theta(k, n) \) (largest root of \( x^3 - (3n - 2k - 5)x^2 + n(2n - 2k - 5)x - 2(n - k - 3)(n - k - 2) \)) is necessary and sufficient except for a small list of explicit exceptions [2406.07132]. ## 6. Methods: Quotient Matrix, Perron-Frobenius, Edge Operations - Quotient matrix techniques and equitable partitions are extensively used to reduce the computation of signless Laplacian spectral radius for extremal graphs or hypergraphs, yielding closed forms for the largest eigenvalue by solving low-degree characteristic polynomials [2209.03120], [1908.04221], [1612.03538], [1807.05955]. - Edge-moving and grafting operations strictly increase the signless Laplacian spectral radius, providing a structural route to prove uniqueness of extremal graphs and hypertrees [1807.05955], [1612.03538], [2305.03280]. - Spectral monotonicity ensures that any deviation from extremal structure strictly decreases spectral radius, anchoring tightness of bounds for matching, extension, tree, and spanning substructure results [2303.16687], [2406.07132]. --- The signless Laplacian spectral radius thus serves as a unifying extremal parameter connecting combinatorial graph theory, spectral methods in hypergraphs, and topological properties of complexes, with explicit extremal constructions and sharp spectral bounds confirmed for many Turán-type, forbidden subgraph, matching, factor, and spanning tree existence problems. For all such scenarios, equality cases are typically realized by highly structured, often join-type or multipartite graphs, supertrees, or complexes, and these bounds subsume classical combinatorial results under a spectral lens. Source: https://www.emergentmind.com/topics/signless-laplacian-spectral-radius