Linear combinations of graph eigenvalues
نویسنده
چکیده
Let 1 (G) : : : n (G) be the eigenvalues of the adjacency matrix of a graph G of order n; and G be the complement of G: Suppose F (G) is a xed linear combination of i (G) ; n i+1 (G) ; i G ; and n i+1 G ; 1 i k: We show that the limit lim n!1 1 n max fF (G) : v (G) = ng always exists. Moreover, the statement remains true if the maximum is taken over some restricted families like Kr-freeor r-partitegraphs. We also show that 29 + p 329 42 n 25 max v(G)=n 1 (G) + 2 (G) 2 p 3 n; answering in the negative a question of Gernert. AMS classi cation: 15A42, 05C50 Keywords: extremal graph eigenvalues, linear combination of eigenvalues, multiplicative property 1 Introduction Our notation is standard (e.g., see [1], [3], and [7]); in particular, all graphs are de ned on the vertex set [n] = f1; : : : ; ng and G stands for the complement of G: We order the eigenvalues of the adjacency matrix of a graph G of order n as 1 (G) : : : n (G) : Suppose k > 0 is a xed integer and 1; : : : ; k; 1; : : : ; k; 1; : : : ; k; 1; : : : ; k; are xed reals. For any graph G of order at least k; let
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