On graphs with smallest eigenvalue at least −3 and their lattices

Graphs and Combinatorics

Rehman

Yang

2019

Self Cite

“…For the case of graphs with smallest eigenvalue at least −3, Koolen, Yang and Yang [24] generalized a result of Hoffman (1977) [21] and showed the following. Theorem 1.3.…”

Section: Introductionmentioning

confidence: 91%

On the Integrability of Strongly Regular Graphs

Graphs and Combinatorics

Rehman

Yang

2019

Self Cite

“…An interesting direction would be to prove a similar result for signed graphs. Koolen, Yang and Yang [59] also introduced (−3)-maximal graphs or maximal graphs with smallest eigenvalue −3. These are connected graphs with smallest eigenvalue at least −3 such any proper connected supergraph has smallest eigenvalue less than −3.…”

Section: The Smallest Eigenvalue Of Signed Graphsmentioning

confidence: 99%

“…Koolen and Munemasa [57] proved that the join between a clique on three vertices and the complement of the McLaughlin graph (see Goethals and Seidel [43] or Inoue [55] for a description) is (−3)-maximal. Woo and Neumaier [80] introduced the notion of Hoffman graphs, which has proved an essential tool in many results involving the smallest eigenvalue of unsigned graphs (see [59]). Perhaps a theory of signed Hoffman graphs is possible as well.…”

Section: The Smallest Eigenvalue Of Signed Graphsmentioning

confidence: 99%

Open problems in the spectral theory of signed graphs

Belardo

Cioabă

Art Discrete Appl. Math.

et al. 2019

Signed graphs are graphs whose edges get a sign +1 or −1 (the signature). Signed graphs can be studied by means of graph matrices extended to signed graphs in a natural way. Recently, the spectra of signed graphs have attracted much attention from graph spectra specialists. One motivation is that the spectral theory of signed graphs elegantly generalizes the spectral theories of unsigned graphs. On the other hand, unsigned graphs do not disappear completely, since their role can be taken by the special case of balanced signed graphs.Therefore, spectral problems defined and studied for unsigned graphs can be considered in terms of signed graphs, and sometimes such generalization shows nice properties which cannot be appreciated in terms of (unsigned) graphs. Here, we survey some general results on the adjacency spectra of signed graphs, and we consider some spectral problems which are inspired from the spectral theory of (unsigned) graphs.

“…Later in 2018, Koolen et al [62] studied the integrability of graphs with smallest eigenvalue at least −3 and proved that:…”

Section: S-integrability Of Graphsmentioning

confidence: 99%

“…[62, Theorem 5.3]). There exists a positive integer κ 4 such that, if G is a graph with smallest eigenvalue at least −3 and minimal valency at least κ 4 , then G is the slim graph of a fat Hoffman graph with smallest eigenvalue at least −3.We now show how to obtain Theorem 2.2 from Theorem 4.8.…”

mentioning

confidence: 99%

Problems on Graphs with Fixed Smallest Eigenvalue

Yang

2020

Algebra Colloq.

Self Cite

For strongly regular graphs with parameters (v, k, a, c) and fixed smallest eigenvalue −λ, Neumaier [5] gave two bounds of c by using algebraic property of strongly regular graphs. We define a new class of regular graphs called sesqui-regular graphs, which are amply regular graphs without parameter a. We prove that if a sesqui-regular graph * J.H. with parameters (v, k, c) has very large k and fixed smallest eigenvalue −λ, then c ≤ λ 2 (λ − 1) or v − k − 1 ≤ (λ−1) 2 4 + 1 holds.