Signless Laplacian Spectral Radius

• The signless Laplacian spectral radius is defined as the largest eigenvalue of the sum of the degree and adjacency matrices, reflecting key structural properties in graphs and hypergraphs.
• Extremal results reveal that uniquely structured join and multipartite graphs achieve sharp spectral bounds in forbidden subgraph, tree, and Turán-type problems.
• Analytical methods such as Perron–Frobenius theory, equitable partitioning, and tensor eigenpair frameworks provide actionable insights for spanning trees, matchings, and spectral inequalities.

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)$0, $Q(G) = D(G) + A(G)$1 the diagonal of degrees, $Q(G) = D(G) + A(G)$2 the adjacency matrix.
• The signless Laplacian spectral radius $Q(G) = D(G) + A(G)$3 is the largest eigenvalue of $Q(G) = D(G) + A(G)$4; equivalently, $Q(G) = D(G) + A(G)$5, and for connected $Q(G) = D(G) + A(G)$6 there is a unique positive unit eigenvector (Perron vector).

For $Q(G) = D(G) + A(G)$7-uniform hypergraphs $Q(G) = D(G) + A(G)$8, the signless Laplacian is a symmetric order-$Q(G) = D(G) + A(G)$9 tensor $D(G)$0, with:

• $D(G)$1 encoding edge incidence (normalized as $D(G)$2 if $D(G)$3), $D(G)$4 diagonal in degree.
• Tensor eigenpair $D(G)$5 satisfies $D(G)$6, with $D(G)$7.
• The spectral radius $D(G)$8 is the maximum modulus among eigenvalues and is attained for a nonnegative principal eigenvector by nonnegative tensor theory (Duan et al., 2018, Duan et al., 2020, Lu et al., 13 Jan 2026).

2. Key Structural Extremal Results and Constructions

Graphs

• For forbidden subgraphs and trees: For graphs omitting any tree on $D(G)$9 vertices, the extremal structure is $A(G)$0, and for trees of size $A(G)$1, the extremal graph is $A(G)$2 (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 $A(G)$3, with the closed formula $A(G)$4 (Chen et al., 2022).
• For $A(G)$5-minor free graphs: The extremal graphs are $A(G)$6 with sharp upper bound for signless Laplacian spectral radius and explicit characterization, particularly for $A(G)$7 (Chen et al., 2019).
• For forbidden cycles or fans ($A(G)$8), the unique maximizer is the complete split graph $A(G)$9 (Zhao et al., 2020), and for forbidden intersecting odd cycles the extremal configuration is also $Q(G)$0 with precise spectral bounds (Chen et al., 2021, Liu et al., 2024).
• For planar graphs, the join construction $Q(G)$1 achieves the maximal signless Laplacian spectral radius among all planar graphs of large order (Yu, 2014).

Hypergraphs

• For $Q(G)$2-uniform supertrees, among all with prescribed edge count $Q(G)$3 and diameter $Q(G)$4, the unique maximizer for signless Laplacian spectral radius is constructed by attaching all excess edges "in the middle" (at $Q(G)$5); 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 (Duan et al., 2018).
• 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 $Q(G)$6-graph for Fano plane-free hypergraphs yields sharp bounds on signless Laplacian spectral radius (Lu et al., 13 Jan 2026).
• For general hypergraphs, the lower bound $Q(G)$7, where $Q(G)$8 is the clique number and $Q(G)$9 the set of allowed edge sizes, is sharp (with equality for the complete $q(G)$0-hypergraph of size $q(G)$1) (Duan et al., 2020).

Simplicial Complexes

• For a pure $q(G)$2-complex free of top $q(G)$3-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 $q(G)$4 (Fan et al., 30 Jul 2025).

3. Spectral Bounds, Inequalities, and Characterizations

• For a graph $q(G)$5 with vertex degrees $q(G)$6, classical and refined bounds include:
  • $q(G)$7 (adjacency spectral radius plus maximal degree) (Hong et al., 2013).
  • Merris-Feng-Yu bound: $q(G)$8, with equality only for regular graphs (Hong et al., 2013, Kong et al., 2016).
  • Bounds in terms of clique number: for connected $q(G)$9 of order $k$0, clique number $k$1, the extremal Turán graph achieves the upper bound (He et al., 2012).
• 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) (Hong et al., 2014).
• For distance signless Laplacian, transmission-regularity characterizes the equality cases (Hong et al., 2013).

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 (Zheng et al., 3 Jul 2025).
  • Analogous results for hypergraphs and simplicial complexes: spectral bounds translate into face-number (count of $k$2-faces) bounds, e.g., for hole-free pure $k$3-dimensional complexes, Turán-type upper bounds on $k$4 follow from spectral extremality (Fan et al., 30 Jul 2025, Lu et al., 13 Jan 2026).

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 (Liu et al., 2020, Zhou et al., 2023, Wang et al., 2024).
  • For $k$5-extendability, cubic and quadratic spectral bounds determine sharp existence conditions; extremal non-$k$6-extendable graphs attain equality in the bound (Zhou et al., 2023).
  • For spanning trees with leaf degree at most $k$7, spectral threshold $k$8 (largest root of $k$9) is necessary and sufficient except for a small list of explicit exceptions (Wang et al., 2024).

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 (Chen et al., 2022, Chen et al., 2019, Kong et al., 2016, Duan et al., 2018).
• Edge-moving and grafting operations strictly increase the signless Laplacian spectral radius, providing a structural route to prove uniqueness of extremal graphs and hypertrees (Duan et al., 2018, Kong et al., 2016, Chen et al., 2023).
• 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 (Zhou et al., 2023, Wang et al., 2024).

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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.