Quadratic forms on graphs with application to minimizing the least eigenvalue of signless Laplacian over bicyclic graphs

Abstract

Given a graph and a vector defined on the graph, a quadratic form is defined on the graph depending on its edges. In order to minimize the quadratic form on trees or unicyclic graphs associated with signless Laplacian, the notion of basic edge set of a graph is introduced, and the behavior of the least eigenvalue and the corresponding eigenvectors is investigated. Using these results a characterization of the unique bicyclic graph whose least eigenvalue attains the minimum among all non-bipartite bicyclic graphs of fixed order is obtained.

The least signless Laplacian eigenvalue of the complements of bicyclic graphs

“…Wang and Fan [8] determined the graph whose the least signless Laplacian eigenvalue is minimum. Yu, Fan and Wang [12] determined the unique graph whose least signless Laplacian eigenvalue attains the minimum among all connected non-bipartite bicyclic graphs. Yu, Guo and Xu [10] determined the unique graph whose least signless Laplacian eigenvalue attains the minimum among all connected non-bipartite graphs with given matching number and edge cover number, respectively.…”

The Least Eigenvalue of the Graphs Whose Complements Are Connected and Have Pendent Paths

“…However, relative to the (signless Laplacian) spec-tral radius, there are few results on the (signless Laplacian) least eigenvalue. Especially, when the structures of graphs are very complex, but the structures of their complements are simple, we naturally think whether we can study the (signless Laplacian) minimum eigenvalue of the graphs from structure of their complements; see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. In this paper, we also study the least eigenvalue of graphs from their complements.…”

On the least signless Laplacian eigenvalue of a non-bipartite connected graph with fixed maximum degree

“…Motivated by this problem, a good deal of attention has been devoted to finding all graphs with the minimal least signless Laplacian eigenvalue among a given class of graphs. For related results, one may refer to [ 5 – 14 ].…”