A conjecture on the eigenvalues of threshold graphs

Abstract We consider a particular class of signed threshold graphs and their eigenvalues. If Ġ is such a threshold graph and Q(Ġ ) is a quotient matrix that arises from the equitable partition of Ġ , then we use a sequence of elementary matrix operations to prove that the matrix Q(Ġ ) – xI (x ∈ ℝ) is row equivalent to a tridiagonal matrix whose determinant is, under certain conditions, of the constant sign. In this way we determine certain intervals in which Ġ has no eigenvalues.

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ON THE PAIRWISE COMPATIBILITY PROPERTY OF SOME SUPERCLASSES OF THRESHOLD GRAPHS

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830913600021 ◽

2013 ◽

Vol 05 (02) ◽

pp. 1360002 ◽

Cited By ~ 7

Author(s):

TIZIANA CALAMONERI ◽

ROSSELLA PETRESCHI ◽

BLERINA SINAIMERI

Keyword(s):

The Other ◽

Real Numbers ◽

Weighted Tree ◽

Positive Edge ◽

Threshold Graphs ◽

Compatibility Graph ◽

Unique Path ◽

Compatibility Property

A graph G is called a pairwise compatibility graph (PCG) if there exists a positive edge weighted tree T and two non-negative real numbers d min and d max such that each leaf lu of T corresponds to a node u ∈ V and there is an edge (u, v) ∈ E if and only if d min ≤ dT (lu, lv) ≤ d max , where dT (lu, lv) is the sum of the weights of the edges on the unique path from lu to lv in T. In this paper we study the relations between the pairwise compatibility property and superclasses of threshold graphs, i.e., graphs where the neighborhoods of any couple of nodes either coincide or are included one into the other. Namely, we prove that some of these superclasses belong to the PCG class. Moreover, we tackle the problem of characterizing the class of graphs that are PCGs of a star, deducing that also these graphs are a generalization of threshold graphs.

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On Universal Threshold Graphs

Combinatorics, Geometry and Probability ◽

10.1017/cbo9780511662034.034 ◽

1997 ◽

pp. 375-392

Author(s):

P.L. Hammer ◽

A.K. Kelmans

Keyword(s):

Threshold Graphs

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Pathwidth and Searching in Parameterized Threshold Graphs

WALCOM: Algorithms and Computation - Lecture Notes in Computer Science ◽

10.1007/978-3-642-11440-3_27 ◽

2010 ◽

pp. 293-304 ◽

Cited By ~ 1

Author(s):

D. Sai Krishna ◽

T. V. Thirumala Reddy ◽

B. Sai Shashank ◽

C. Pandu Rangan

Keyword(s):

Threshold Graphs

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Oriented incidence energy and threshold graphs

Filomat ◽

10.2298/fil1102001s ◽

2011 ◽

Vol 25 (2) ◽

pp. 1-8 ◽

Cited By ~ 11

Author(s):

Dragan Stevanovic

Keyword(s):

Incidence Matrix ◽

Oriented Graph ◽

Singular Values ◽

Simple Graph ◽

Arbitrary Orientation ◽

Threshold Graphs

Let G be a simple graph with n vertices and m edges. Let edges of G be given an arbitrary orientation, and let Q be the vertex-edge incidence matrix of such oriented graph. The oriented incidence energy of G is then the sum of singular values of Q. We show that for any n?9, there exists at least ([n/9]/2)+1 distinct pairs of graphs on n vertices having equal oriented incidence energy.

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Q-borderenergetic threshold graphs

Ciência e Natura ◽

10.5902/2179460x39755 ◽

2020 ◽

Vol 42 ◽

pp. e91

Author(s):

João Roberto Lazzarin ◽

Oscar Franscisco Másquez Sosa ◽

Fernando Colman Tura

Keyword(s):

Complete Graph ◽

Threshold Graphs ◽

Laplacian Energy ◽

Signless Laplacian

A graph G is said to be borderenergetic (L-borderenergetic, respectively) if its energy (Laplacian energy, respectively) equals the energy (Laplacian energy, respectively) of the complete graph. Recently, this concept was extend to signless Laplacian energy (see Tao, Q., Hou, Y. (2018). Q-borderenergetic graphs. AKCE International Journal of Graphs and Combinatorics). A graph G is called Q-borderenergetic if its signless Laplacian energy is the same of the complete graph Kn; i.e., QE(G) = QE(Kn) = 2n - 2: In this paper, we investigate Q-borderenergetic graphs on the class of threshold graphs. For a family of threshold graphs of order n = 100; we find out exactly 13 graphs such that QE(G) = 2n- 2:

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Note on chromatic polynomials of the threshold graphs

Electronic Journal of Graph Theory and Applications ◽

10.5614/ejgta.2019.7.2.2 ◽

2019 ◽

Vol 7 (2) ◽

pp. 217-224

Author(s):

Noureddine Chikh ◽

◽

Miloud Mihoubi ◽

Keyword(s):

Chromatic Polynomials ◽

Threshold Graphs

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On $xD$-Generalizations of Stirling Numbers and Lah Numbers via Graphs and Rooks

The Electronic Journal of Combinatorics ◽

10.37236/6699 ◽

2017 ◽

Vol 24 (2) ◽

Cited By ~ 2

Author(s):

Sen-Peng Eu ◽

Tung-Shan Fu ◽

Yu-Chang Liang ◽

Tsai-Lien Wong

Keyword(s):

Weyl Algebra ◽

Stirling Numbers ◽

Stirling Number ◽

Combinatorial Structures ◽

Normal Ordering ◽

Rook Placements ◽

Threshold Graphs ◽

Combinatorial Interpretations ◽

Ferrers Boards

This paper studies the generalizations of the Stirling numbers of both kinds and the Lah numbers in association with the normal ordering problem in the Weyl algebra $W=\langle x,D|Dx-xD=1\rangle$. Any word $\omega\in W$ with $m$ $x$'s and $n$ $D$'s can be expressed in the normally ordered form $\omega=x^{m-n}\sum_{k\ge 0} {{\omega}\brace {k}} x^{k}D^{k}$, where ${{\omega}\brace {k}}$ is known as the Stirling number of the second kind for the word $\omega$. This study considers the expansions of restricted words $\omega$ in $W$ over the sequences $\{(xD)^{k}\}_{k\ge 0}$ and $\{xD^{k}x^{k-1}\}_{k\ge 0}$. Interestingly, the coefficients in individual expansions turn out to be generalizations of the Stirling numbers of the first kind and the Lah numbers. The coefficients will be determined through enumerations of some combinatorial structures linked to the words $\omega$, involving decreasing forest decompositions of quasi-threshold graphs and non-attacking rook placements on Ferrers boards. Extended to $q$-analogues, weighted refinements of the combinatorial interpretations are also investigated for words in the $q$-deformed Weyl algebra.

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