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Abstract. In this work we present eigenvalue interlacing results for the adjacency matrix A, the standard Laplacian matrix L, the normalized Laplacian matrix L and the signless Laplacian matrix Q associated with the edge subdivision operation. In addition, we show an eigenvalue interlacing result associated with the vertex contraction operation for the signless Laplacian matrix Q.

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Abstract. Let G be a simple graph with n vertices, andIn this work, we present upper bounds for S + k (G) for K 2,s+1 -free graphs, for even cycle-free graphs and for odd cycle-free graphs.

Resumo. In this work we develop results to contribute to the study of the Brouwer conjecture through an argument of contradiction. Specifically, if the Brouwer conjecture is not valid for a graph and this graph respects some conditions, we show that the conjecture is also not valid for the graphs obtained by deleting an edge or vertex. In this way, we can recursively delete certain vertices and edges of the original graph until the resulting graph is in a family for which the conjecture is proven, a contradiction.