Signless Laplacian characterization of cones over disjoint unions of cycles, edges and isolated vertices

Two graphs are said to be Q -cospectral if they share the same signless Laplacian spectrum. A simple graph is said to be determined by its signless Laplacian spectrum (abbreviated as DQS) if there exists no other non-isomorphic simple graph with the same signless Laplacian spectrum. In this paper, we establish the following results: (1) Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>G</m:mi> <m:mo>≅</m:mo> <m:msub> <m:mrow> <m:mi>K</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>∨</m:mo> <m:mfenced close=")" open="("> <m:mrow> <m:msub> <m:mrow> <m:mi>C</m:mi> </m:mrow> <m:mrow> <m:mi>k</m:mi> </m:mrow> </m:msub> <m:mo>∪</m:mo> <m:mi>q</m:mi> <m:msub> <m:mrow> <m:mi>K</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:mo>∪</m:mo> <m:mi>s</m:mi> <m:msub> <m:mrow> <m:mi>K</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> </m:mfenced> <m:mo>,</m:mo> </m:math> G K₁ (Cₖ∪ qK₂∪ sK₁), with q , s ≥ 1, k ≥ 4, and at least 21 vertices. If k is odd, then G is DQS. Moreover, if k is even and F is Q -cospectral with G , then <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll"> <m:mi>F</m:mi> <m:mo>≅</m:mo> <m:mi>G</m:mi> <m:mtext> </m:mtext> <m:mtext>or</m:mtext> <m:mtext> </m:mtext> <m:mi>F</m:mi> <m:mo>≅</m:mo> <m:msub> <m:mrow> <m:mi>K</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>∨</m:mo> <m:mfenced close=")" open="("> <m:mrow> <m:msub> <m:mrow> <m:mi>C</m:mi> </m:mrow> <m:mrow> <m:mn>4</m:mn> </m:mrow> </m:msub> <m:mo>∪</m:mo> <m:msub> <m:mrow> <m:mi>P</m:mi> </m:mrow> <m:mrow> <m:mi>k</m:mi> <m:mo>−</m:mo> <m:mn>3</m:mn> </m:mrow> </m:msub> <m:mo>∪</m:mo> <m:msub> <m:mrow> <m:mi>P</m:mi> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:msub> <m:mo>∪</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>q</m:mi> <m:mo>−</m:mo> <m:mn>2</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:msub> <m:mrow> <m:mi>K</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:mo>∪</m:mo> <m:mi>s</m:mi> <m:msub> <m:mrow> <m:mi>K</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> </m:mfenced> <m:mo>.</m:mo> </m:math> F G or F K₁ (C₄∪ Pₖ₋₃∪ P₃∪ (q-2)K₂∪ sK₁). (2) Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>G</m:mi> <m:mo>≅</m:mo> <m:msub> <m:mrow> <m:mi>K</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>∨</m:mo> <m:mrow>