On the Spectrum of Threshold Graphs

Abstract

The antiregular connected graph onrvertices is defined as the connected graph whose vertex degrees take the values ofr−1distinct positive integers. We explore the spectrum of its adjacency matrix and show common properties with those of connected threshold graphs, having an equitable partition with a minimal numberrof parts. Structural and combinatorial properties can be deduced for related classes of graphs and in particular for the minimal configurations in the class of singular graphs.

Some new considerations about double nested graphs

“…In [19] it was proved that all eigenvalues of any threshold graph other than 0, 1 are main. However the analogous statement in the case of DNGs is not true.…”

Eigenvalue location in threshold graphs

“…As far as threshold graphs, spectral properties were studied by Sciriha and Farrugia in [14]. They showed that all eigenvalues, other than −1 or 0, are main.…”

“…Spectral properties of co-graphs, which contain the class of threshold graphs, were studied in [2]. Threshold graphs were studied in [15,16] under the name nested split graphs.…”

“…If n ≡ 2 mod 3, an additional graph, namely G(n − n 3 , n 3 ), also has eigenvalue λ min,n . The work in [15] is related to ours, however this paper seeks bounds for the largest eigenvalue of a given threshold graph. Our paper finds the exact lower bound of all degree n threshold graphs, and the graph(s) which attain this lower bound.…”

Grafos Threshold Equienergéticos

“…Embora na prática, na maioria dos casos, issoé não trivial, pois não possuímos método algébrico para encontrar raízes de polinômios de grau maior do que 5, a classe de grafos threshold possui uma particularidade. Se Aé a matriz de adjacência de um grafo threshold G, o polinômio característico de G pode ser obtido através de uma redução da matriz A. Aqui será apresentado a idéia geral dessa redução, para mais detalhes, consultar [8]. Seja G um grafo threshold com sequência binária (…”