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Spectral Theory of Signed Graphs and Digraphs

Nematollahi, Mohammad Ali | 2019

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  1. Type of Document: Ph.D. Dissertation
  2. Language: Farsi
  3. Document No: 52429 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Akbari, Saieed
  7. Abstract:
  8. A signed graph is a pair like (G; ), where G is the underlying graph and : E(G) ! f1; +1g is a sign function on the edges of G. Here, we study the spectral determination problem for signed n-cycles (Cn; ) with respect to the adjacency spectrum and the Laplacian spectrum. In particular we prove that signed odd cycles and unbalanced even cycles are uniquely determined by their Laplacian spectrums, but balanced even cycles are not, and we find all L-cospectral mates for them. Moreover, signed odd cycles are uniquely determined by their spectrums but the signed even cycles, (except (C4;) and (C4; +)), are not and we find almost all cospectral mates for them. A mixed graph is obtained from a graph by orienting some of its edges. The Hermitian adjacency matrix of a mixed graph with the vertex set fv1; : : : ; vng, is the matrix H = [hij ]nn, where hij = hji = i if there is a directed edge from vi to vj , hij = 1 if there exists an undirected edge between vi and vj , and hij = 0 otherwise. The Hermitian spectrum of a mixed graph is defined to be the spectrum of its Hermitian adjacency matrix. First, we show that up to switching equivalence, of each order, there exist three types of mixed cycles and we determine their Hermitian spectrums.Hence, the Hermitian spectrums of all mixed cycles are obtained.We characterize all mixed paths and mixed cycles which are determined by their Hermitian spectrums and for those that are not, we present some Cospectral mates. Finally, we solve the similar problem for mixed cycles. Next, we study a conjecture of Haemers on the Seidel energy of graphs. Indeed, from the point of view of the theory of signed graphs, the conjecture has the following formation: If we negate some arbitrary edges of (Kn; +), then the energy of the resulting graph is at least 2n 2, the energy of (Kn; +). We prove this conjecture completely
  9. Keywords:
  10. Spectrums ; Laplacian Spectrum ; Hermitian Spectrum ; Mixed Graph ; Siedel Energy ; Signed Graph ; Directed Graph

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