Fractional Decompositions and the Smallest-eigenvalue Separation

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 [5] about the smallest eigenvalue of non-bipartite distance-regular graphs. arXiv:1907.08447v1 [math.CO]

Some observations on the smallest adjacency eigenvalue of a graph

“…In Section 2, we use weighted graph decompositions of the edge set of a graph to bound the spectrum of a graph from below. Our results are similar and have been obtained independently from the recent work of Knox and Mohar [27].…”

“…If S = 2 uv∈M x u x v , then S ≤ 4(1−S) (λ−1) 2 which implies that S ≤ 4 4+(λ−1) 2 . Plugging this into (25), we get that λ ≥ −2 − 4 4 + (λ − 1) 2 (27) which implies that λ 3 − λ + 14 ≥ 0. Hence, λ ≥ θ ≈ −2.272.…”

A spectral bound for vertex-transitive graphs and their spanning subgraphs

“…Using fractional decomposition of graphs, Knox-Mohar have shown that the smallest eigenvalue of the normalized adjacency operator of a undirected, regular, nonbipartite, distance-regular graph with odd girth g is greater than or equal to 1 − cos π g [19,Corollary 5], generalizing and strengthening a result of Qiao-Jing-Koolen [26, Theorem 1].…”

A lower bound for the smallest eigenvalue of a graph and an application to the associahedron graph

“…Aharoni, Alon, and Berger [2] obtained a lower bound for the smallest eigenvalue of a regular graph where each vertex is contained in many triangles (see also [8]). Knox and Mohar [21] obtained a lower bound for the smallest eigenvalue using graph decompositions and their work leads to a simpler proof of a result of Qiao, Jing, and Koolen [33] on the smallest eigenvalue of a distance-regular graph.…”