In this paper we characterize all graphs with exactly two non-negative eigenvalues. As a consequence we obtain all graphs G such that λ 3 (G) < 0, where λ 3 (G) is the third largest eigenvalue of G.
Let G be a simple graph on n vertices. An independent set in a graph is a set of pairwise non-adjacent vertices. The independence polynomial of G is the polynomial I(G, x) = n k=0 s(G, k)x k , where s(G, k) is the number of independent sets of G with size k and s(G, 0) = 1. A unicyclic graph is a graph containing exactly one cycle. Let C n be the cycle on n vertices. In this paper we study the independence polynomial of unicyclic graphs. We show that among all connected unicyclic graphs G on n vertices (except two of them), I(G, t) > I(C n , t) for sufficiently large t. Finally for every n ≥ 3 we find all connected graphs H such that I(H, x) = I(C n , x).
Let $G$ be a graph of order $n$ with signless Laplacian eigenvalues $q_1, \ldots,q_n$ and Laplacian eigenvalues $\mu_1,\ldots,\mu_n$. It is proved that for any real number $\alpha$ with $0 < \alpha\leq1$ or $2\leq\alpha < 3$, the inequality $q_1^\alpha+\cdots+ q_n^\alpha\geq \mu_1^\alpha+\cdots+\mu_n^\alpha$ holds, and for any real number $\beta$ with $1 < \beta < 2$, the inequality $q_1^\beta+\cdots+ q_n^\beta\le \mu_1^\beta+\cdots+\mu_n^\beta$ holds. In both inequalities, the equality is attained (for $\alpha \notin \{1,2\}$) if and only if $G$ is bipartite.
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