A Lower Bound for the Spectral Radius of Graphs with Fixed Diameter

Abstract

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The Maximum Spectral Radius of Non-Bipartite Graphs Forbidding Short Odd Cycles

“…There is a rich history on the study of bounding the eigenvalues of a graph in terms of various parameters; see [2] for eigenvalues and expanders, [11,15] for eigenvalues and diameters, [21] for spectral radius and genus, [3] for spectral radius and cut vertices, [12,34] for regularity and eigenvalues, [13,33] for non-regularity and spectral radius, [7] for spectral radius and cliques, [4,52] for chromatic number and eigenvalues, [35,17,40] for independence number and eigenvalues, [14,46] for matching, edge-connectivity and eigenvalues, [18] for spanning trees and eigenvalues, [48,30] for eigenvalues of outerplanar and planar graphs, and [49] for the Colin de Verdière parameter, excluded minors and the spectral radius.…”

Graphs with the minimum spectral radius for given independence number

“…The results for giving diameter is only determined for some special values. For examples, the graph G of the minimum spectral radius with small diameter diam(G) ∈ {1, 2, 3, 4} are determined in [1,3], with large diameter diam(G) ∈ {n − 1, n − 2, n − 3, ⌊n/2⌋} in [7], diam(G) = n − 4 in [21], simultaneously, diam(G) ∈ {n − 4, n − 5} in [2], and diam(G) ∈ {n − 6, n − 7, n − 8} ∪ [ n 2 , 2n− 3 3 ] in [14]. In general, characterizing the graphs of the minimum spectral radius is still an open problem [17].…”