Signless Laplacian extremal graphs for complete bipartite graphs plus an edge

Determine the signless Laplacian spectral extremal graphs among all n-vertex graphs that are free of the graph obtained from the complete bipartite graph K_{s,t} by adding an edge within the part of size s, proving that every extremal graph belongs to either the family of joins of K_{s-1} with a nearly (t-1)-regular triangle-free graph or the family of joins of an independent set of size t-1 with a nearly (t-1)-regular triangle-free graph.

Background

The paper considers the signless Laplacian spectral Turán problem for color-critical forbidden graphs. Its main theorem treats color-critical graphs of chromatic number at least four, whereas color-critical graphs of chromatic number three remain outside the scope of the established method. The graph obtained by adding an edge to one part of K_{s,t} is introduced as a natural candidate for studying this unresolved chromatic-number-three case.

For fixed integers 2 ≤ s ≤ t, the authors define two candidate extremal families: joins of K_{s-1} with nearly (t-1)-regular triangle-free graphs, and joins of an independent set of size t-1 with nearly (t-1)-regular triangle-free graphs. The conjecture asserts that every n-vertex forbidden-subgraph-free graph with maximum signless Laplacian spectral radius belongs to one of these two families. The paper notes that the s = 2 case has been studied, while the case s ≥ 3 may exhibit different behavior.

References

In this case, we propose the following conjecture. Let 2 ≤ s ≤t be positive integers. We define the following families: . Let En,s,t be the family of graphs that are the join of a clique Ks-1 and a (nearly) (t-1)-regular triangle-free graph of order n - s + 1. . Let yn,t be the family of graphs that are the join of an independent set It-1 and a (nearly) (t - 1)-regular triangle-free graph of order n - t + 1. Clearly, all graphs in both En,s,t and yn,t are Kgt-free. We point out that such a regular graph in the above may not exist, but we can always choose a nearly regular graph. It is well-known that if t is even and n is odd, then there exist nearly (t-1)-regular graphs of order n whose degree sequence is (t-1, ... ,t-1,t - 2). Otherwise, there exist (t - 1)-regular graphs of order n.

Conjecture 5.1. Let 2 ≤ s ≤t and n ≥s+t. IfG is an n-vertex Kgt-free graph with the maximal signless Laplacian spectral radius, then G is a member of either Ln,s,t or yn,t.

— The signless Laplacian spectral Turán problems for color-critical graphs  (2504.07852 - Zheng et al., 10 Apr 2025) in Conjecture 5.1, Section 5.1, “Forbidding complete bipartite graphs plus an edge”