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Graphs whose spectrum determined by non-constant coefficients

Akbari, S ; Sharif University of Technology

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  1. Type of Document: Article
  2. DOI: 10.1016/j.endm.2013.11.007
  3. Abstract:
  4. Let G be a graph and M be a matrix associated with G whose characteristic polynomial is M(G,x)=∑i=0nαi(G)xn-i. We say that the spectrum of G is determined by non-constant coefficients (simply M-SDNC), if for any graph H with ai(H)=ai(G), 0≤i≤n-1, then Spec(G)=Spec(H) (if M is the adjacency matrix or the Laplacian matrix of G, then G is called an A-SDNC graph or L-SDNC graph). In this paper, we study some properties of graphs which are A-SDNC or L-SDNC. Among other results, we prove that the path of order at least five is L-SDNC and moreover stars of order at least five are both A-SDNC and L-SDNC. Furthermore, we construct infinitely many trees which are not A-SDNC graphs. More precisely, we show that there are infinitely many pairs (T, T') of trees such that A(T, x)-A(T', x)=-1
  5. Keywords:
  6. Adjacency matrix of a graph ; Characteristic polynomial of a graph ; Laplacian matrix of a graph
  7. Source: Electronic Notes in Discrete Mathematics ; Vol. 45 , 2014 , pp. 29-34 ; ISSN: 15710653
  8. URL: http://www.sciencedirect.com/science/article/pii/S1571065313002874