Abstract
Let G be an n-vertex (n ? 3) simple graph embeddable on a surface of Euler genus (the number of crosscaps plus twice the number of handles). In this paper, we present upper bounds for the signless Laplacian spectral radius of planar graphs, outerplanar graphs and Halin graphs, respectively, in terms of order and maximum degree. We also demonstrate that our bounds are sometimes better than known ones. For outerplanar graphs without internal triangles, we determine the extremal graphs with the maximum and minimum signless Laplacian spectral radii.
Highlights
Spectral graph theory is a fast growing branch of algebraic graph theory
Cvetkovicand Simicdefined in a series of three papers [6, 7, 8], entitled ”Towards a spectral theory of graphs based on the signless Laplacian”, the fundamentals of the spectral theory of graphs based on the signless Laplacian
The following results were obtained recently on the signless Laplacian spectral radius of graphs embedded on surfaces
Summary
Spectral graph theory is a fast growing branch of algebraic graph theory. Within spectral graph theory, studying the properties of a graph using its signless Laplacian became very dynamic area of research in past decade. Signless Laplacian matrix, spectral radius, Euler genus, outerplanar graph, Halin graph. Cao and Vince [3], independently Boots and Royle [2] by using computer studies, conjectured that the planar graph of a given order with largest spectral radius is K2 Pn−2, the join of K2 and Pn−2 It is noted in [2] that the conjecture is not true for n = 7 and 8, but suggested it was true for all n ≥ 9. The following results were obtained recently on the signless Laplacian spectral radius of graphs embedded on surfaces. We present several new upper bounds for the signless Laplacian spectral radius of planar graphs, outerplanar graphs and Halin graphs, in terms of order and maximum degree. For a special class of outerplanar graphs, we determine the extremal graphs with the maximal and minimal signless Laplacian spectral radii
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