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Signless Laplacian Spectra of Graphs

Kianizad, Mosayeb | 2021

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 54414 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Akbari, Saeed
  7. Abstract:
  8. Let G be a graph of order n. The signless Laplacian matrix or Q-matrix of G is Q(G)=D(G)+A(G), where A(G) is the adjacency matrix of G and D(G) is diagonal degree matrix of G. The signless Laplacian characteristic polynomial or Q-polinomial of G is QG(x)=det(xI-Q(G)) and its roots are called signless Laplacian eigenvalues or Q-eigenvalues of G. If R is vertex-degree incidence matrix of G, then Q(G)=RRt. So Q(G) is a positive semi-definite matrix, i.e. its eigenvalues are none-negative. Let q1(G)≥q2(G)≥…≥qn(G) denote the signless Laplacian eigenvalues of G. A theory in which graphs are studied by means of eigenvalues of Q(G) is called signless Laplaciian theory or Q-theory.In this research, we study a spectral theory of graphs based on signless Laplacian Q(G). This research is organized as follows. At first, we present some notions from Graph theory and Linear Algebra. In Section 2 using of Cvetković report we survey basic properties of Q-eigenvalues and compare it with spectral theories based on adjacency and Laplacian matrices. Section 3 contains treating problems within Q-theory. In Section 4 we study a problem of Nordhaus-Gaddum type, and discuss the bounds on q2(G)+q2(G̅). In Section 5 we state some conjectures on signless Laplacian eigenvalues of graphs
  9. Keywords:
  10. Graph Spectrum ; Signless Laplacian ; Graphs ; Laplacian Eigen Values ; Nordhause-Gaddum Type Inequality ; Q-Eigenvalue

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