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Enostavnemu grafu $G$ z $n$ vozlišči definiramo matriko sosednosti in Laplaceovo matriko. Obe imata realne lastne vrednosti. Lastne vrednosti matrike sosednosti označimo s $theta_1(G) geq cdots geq theta_n(G)$, lastne vrednosti Laplaceove matrike pa z $lambda_1(G) geq cdots geq lambda_n(G) = 0$. V delu študiramo neenakosti Nordhaus-Gaddumovega tipa za lastne vrednosti matrike sosednosti in Laplaceove matrike. To so omejitve na vsote oblik $theta_i(G) + theta_i(overline{G})$ in $lambda_j(G) + lambda_j(overline{G})$ za določene vrednosti indeksov $i$ in $j$, pri čemer je $overline{G}$ komplement grafa $G$. Posebej se osredotočimo na preučevanje vsot za najmanjšo lastno vrednost matrike sosednosti in največji dve lastni vrednosti Laplaceove matrike. For a simple graph $G$ of order $n$, we define its adjacency matrix and Laplacian matrix. Both have real eigenvalues. Let $theta_1(G) geq cdots geq theta_n(G)$ be the eigenvalues of the adjacency matrix and $lambda_1(G) geq cdots geq lambda_n(G) = 0$ the eigenvalues of the Laplacian matrix of graph $G$. We study Nordhaus-Gaddum type inequalities for the eigenvalues of these two matrices. These are upper and lower bounds for sums of the forms $theta_i(G) + theta_i(overline{G})$ and $lambda_j(G) + lambda_j(overline{G})$, where $overline{G}$ denotes the graph complement of $G$. The focus of this work is on the sums for the smallest eigenvalue of the adjacency matrix and the largest two eigenvalues of the Laplacian matrix. |
