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On the sum of the \(k\) largest eigenvalues of graphs and maximal energy of bipartite graphs. (English) Zbl 1411.05156
Summary: Let \(G\) be a graph of order \(n\). Also let \(\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_n\) be the eigenvalues of graph \(G\). In this paper, we present the following upper bound on the sum of the \(k\) \((\leq n)\) largest eigenvalues of \(G\) in terms of the order \(n\) and negative inertia \(\theta\) (the number of negative eigenvalues): \(\sum_{i = 1}^k \lambda_i \leq \frac{n}{2(\theta + 1)}(\theta + \sqrt{\theta(k \theta + k - 1)})\) with equality holding if and only if \(G \cong n K_1\) or \(G \cong \overline{p K}_t\)\((n = p t, p \geq 2)\) with \(k = 1\) (where \(\overline{p K}_t\) is the complete \(p\)-partite graph of \(p\) \(t\) vertices with all partition sets having size \(t\)). Moreover, we obtain a sharp upper bound on the energy of bipartite graphs with given order \(n\) and rank \(r\). We also propose an open problem as follows: Characterize all connected \(d\)-regular bipartite graphs of order \(n\) with 5 distinct eigenvalues and rank \(r = \frac{2 n^2}{(4 d - n)^2}\), where \(d > \frac{n}{4}\). Finally we give a partial solution to this problem.

MSC:
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
Software:
SageMath
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