Abstract
A new method is introduced for bounding the separation between the value of $-k$ and the smallest eigenvalue of a non-bipartite $k$-regular graph. The method is based on fractional decompositions of graphs. As a consequence we obtain a very short proof of a generalization and strengthening of a recent result of Qiao, Jing, and Koolen [Electronic J. Combin. 26(2) (2019), #P2.41] about the smallest eigenvalue of non-bipartite distance-regular graphs.
Highlights
Let us consider a connected graph of order n and its adjacency matrix A(G)
We speak of the eigenvalues the graph G, by which we mean the eigenvalues of A(G)
It follows by the Perron-Frobenius Theorem that |λn(G)| = λ1(G) if and only if the graph G is bipartite
Summary
Let us consider a connected graph of order n and its adjacency matrix A(G). We speak of the eigenvalues the graph G, by which we mean the eigenvalues of A(G). Theorem 1 was used in [5] to classify all non-bipartite distance-regular graphs of diameters D = 4 and D = 5 that have δ(G) k/D, continuing on their earlier result in [4] which was used for diameter D = 3 and was based on a similar spectral lemma bounding δ(G). It is mentioned in [5, Remark 1.2] that ε(g) → 0 as g → ∞, and odd cycles show that ε(g) cos2(.
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