Investigation on spectrum of the adjacency matrix and Laplacian matrix of graph Gl
نویسندگان
چکیده
Let Gl be the graph obtained from Kl by adhering the root of isomorphic trees T to every vertex of Kl, and dk−j+1 be the degree of vertices in the level j. In this paper we study the spectrum of the adjacency matrix A(Gl) and the Laplacian matrix L(Gl) for all positive integer l, and give some results about the spectrum of the adjacency matrix A(Gl) and the Laplacian matrix L(Gl). By using these results, an upper bound for the largest eigenvalue of the adjacency matrix A(Gl) is obtained: λ1(A(Gl)) < max{ max 2≤j≤k−2 { √ dj − 1 + √ dj+1 − 1}, √ dk−1 − 1 + √ dk − l + 1, √ dk − l + 1 + l − 1}, and an upper bound for the largest eigenvalue of the Laplacian matrix L(Gl) is also obtained: μ1(L(Gl)) < max { max 2≤j≤k−2 { √ dj − 1 + dj + √ dj+1 − 1}, √ dk−1 − 1 + dk−1 + √ dk − l + 1, √ dk − l + 1 + dk + 1 } . Key–Words: Adjacency matrix, Laplacian matrix, complete graph, spectrum
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