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Some notes on the threshold graphs
Abstract
In this paper we consider threshold graphs (also called nested split graphs) and investigate some invariants of these graphs which can be of interest in bounding the largest eigenvalue of some graph spectra.
... The objective of the work is to specify the fuzzy systems with the addition of the interval theory in its components. A fuzzy set A on a set X is characterized by a mapping m : X → [0] [1], called the membership function. We shall denote a fuzzy set as A = (X, m). ...
... Though it is very young, it has been growing fast and has numerous applications in various fields. A fuzzy graph [6] ξ = (V, σ, µ) is a non empty crisp set V together with a pair of functions σ : V → [0] [1] and µ : V × V → [0] [1] such that for all x, y, µ(x, y) ≤ σ(x) ∩ σ(y). So µ is a fuzzy relation on σ. ...
... Though it is very young, it has been growing fast and has numerous applications in various fields. A fuzzy graph [6] ξ = (V, σ, µ) is a non empty crisp set V together with a pair of functions σ : V → [0] [1] and µ : V × V → [0] [1] such that for all x, y, µ(x, y) ≤ σ(x) ∩ σ(y). So µ is a fuzzy relation on σ. ...
In this paper, fuzzy threshold graphs, fuzzy alternating -cycles, threshold dimension of fuzzy graphs and fuzzy Ferrers digraphs are defined. We show that fuzzy threshold graphs are fuzzy split graphs. Forbidden configurations of fuzzy Ferrers digraph are described. Also some basic theorems related to the stated graphs have been presented.
... is it true that for any α, 0 < α < 1 2 , for sufficiently large n, H n,k is the unique graph that maximizes the A α -index over H(n, n + k)? ...
... Clearly, there exist positive integers i 0 , i 1 , . . . , i h with 1 ≤ i 0 < i 1 ...
... Furthermore, Peled and Mahadev introduced threshold graphs and related topics on it Peled and Mahadev (1995). Notes on threshold graphs had been introduced by Andelic and Simic (2010). Later on, Samanta and Pal introduced a fuzzy threshold graph (Samanta and Pal 2011). ...
... An application is also given at the last part of the paper (Table 2). (Chvatal and Hammer 1973) Studied set-packing problems on threshold graphs Ordman (1985) Established threshold coverings and resource allocation problem Peled and Mahadev (1995) Studied threshold graphs and related topics on it Tao et al. (2008) Introduced image thresholding Andelic and Simic (2010) Presented notes on threshold graphs Samanta and Pal (2011) Introduced fuzzy threshold graph Pramanik et al. (2016) Proposed interval-valued fuzzy threshold graph Yang and Mao (2019) Studied intuitionistic fuzzy threshold graphs This paper Introduced m-polar fuzzy threshold graph ...
The threshold graph is a well studied topic. But, the fuzzy threshold graph is defined recently and investigated many properties. In a fuzzy threshold graph, only one threshold is considered for every vertex and edge. In a m-polar fuzzy graph, each vertex and each edge has a m number of membership values. So, defining a threshold graph for m-polar fuzzy graph is not easy and needs some new ideas. By considering m thresholds each for each component out of m components of membership values of a vertex or edge, we defined m-polar fuzzy threshold graph (mPFTG). Many interesting properties are also presented on mPFTG. The mPFTG is also decomposed in a very unique way and this decomposition generates m different fuzzy threshold graphs and the properties of decomposed graphs are explained. The important parameters viz. m-polar fuzzy threshold dimension and mPF threshold partition number are also defined and investigated thoroughly. Finally, a real-life application using mPFTG on a resource power controlling system is presented.
... In 2009, Diaconis [2] study the limit theory of large threshold graphs and apply this to a variety of models for random threshold graphs. Then Andelic (2010) [3], investigate some invariants of threshold graph. Jacobs, et all [4] (2013) investigating the eigenvalue location of threshold graph. ...
... In 2009, Diaconis [2] study the limit theory of large threshold graphs and apply this to a variety of models for random threshold graphs. Then Andelic (2010) [3], investigate some invariants of threshold graph. Jacobs, et all [4] (2013) investigating the eigenvalue location of threshold graph. ...
... The next significant result can be found in the same reference and states that a graph which maximizes the spectral radius within ( , ) belongs to the class of the so-called threshold graphs (also known as nested split graphs). The definition can be found in any of Anđelić and Simić (2010), Simić et al. (2004), Stanić (2015), Stevanović (2015a); both names describe the structure of graphs in question. (Again, there is no similar class that would help in the study of our problem.) ...
... Let T n be an antiregular graph of order n. Then T n is a threshold graph that is generated either by the binary sequence (01)(01) · · · (01) k if n = 2k or by the binary sequence (0 2 1)(01) · · · (01) k if n = 2k + 1. (For more details on the generating procedure, spectral and structural properties of threshold graphs, the reader is referred to [2,5]). According to the vertex ordering, where the vertices are ordered according to their vertex degrees in non-increasing order, the Laplacian matrix of T n is ...
In this paper, we provide a new family of tridiagonal matrices whose eigenvalues are perfect squares. This result motivates the computation of the spectrum of a particular antibidiagonal matrix. As an application, we consider the Laplacian controllability of a particular subclass of chain graphs known as half graphs.
... Tao et al. [10] introduced image thresholding by using graph cuts. Some notes on TGs had been studied by Andelic and Simic [11]. The concept of FTGs was first initiated by Samanta and Pal [1]. ...
In this study, a novel concept of picture fuzzy threshold graph (PFTG) is introduced. It has been shown that PFTGs are free from alternating 4-cycle and it can be constructed by repeatedly adding a dominating or an isolated node. Several properties about PFTGs are discussed and obtained the results that every picture fuzzy graph (PFG) is equivalent to a PFTG under certain conditions. Also, the underlying crisp graph (UCG) of PFTG is a split graph (SG), and conversely, a given SG can be applied to constitute a PFTG. A PFTG can be decomposed in a unique way and it generates three distinct fuzzy threshold graphs (FTGs). Furthermore, two important parameters i.e., picture fuzzy (PF) threshold dimension (TD) and PF partition number (PN) of PFGs are defined. Several properties on TD and PN have also been discussed. Lastly, an application of these developed results are presented in controlling medicine resources.
... After that, Peled and Mahadev (1995) explored T G s and related subjects. Andelic and Simic (2010) described T G s in some detail. Smanta and Pal (2011) were the first to discover T G s under fuzzy environment. ...
The aggregation feature, decision-making skills and operational characteristics of multi-purpose Dombi operators make them a highly adaptable tool for compiling the imprecise information. This study exploits the generalized structure of Dombi operators and significant characteristics of picture fuzzy sets (PFSs) to extend the theory of fuzzy graph by presenting the premium concept of picture Dombi fuzzy threshold graphs (PDFTGs). We prove that PDFTGs do not induce picture Dombi fuzzy alternating (PDFA) 4-cycle as induced subgraph, and these graphs can be constructed periodically by adding an isolated or dominant vertex to a single vertex graph. We demonstrate that PDFTGs are triangulated graphs. We show that the crisp graph of PDFTG is a split graph (SG). Further, we illustrate the notion of threshold dimension and threshold partition number of picture Dombi fuzzy graphs (PDFGs). Moreover, we present some fundamental results related to threshold dimension and threshold partition number with the appropriate illustration. Finally, we discuss the implementation of PDFTGs in the distribution of coal resources.
... Afterwards, Peled and Mahadev [29] discussed TG s and related topics on them. Andelic and Simic [8] gave some characterizations of TG s . After that, fuzzy threshold graphs (FTG s ) were first discovered by Smanta and Pal [33]. ...
The purpose of this article is to present a novel idea of complex Pythagorean fuzzy threshold graphs \((\mathcal {CPFTG}_{s})\). We introduce the relation between vertex cardinality and threshold values of a \(\mathcal {CPFTG}\). We propose that \(\mathcal {CPFTG}_{s}\) are free from alternating \(4-cycle\) and these graphs can be built up repeatedly adding an isolated or a dominating vertex. We present that the crisp graph of \(\mathcal {CPFTG}\) is a split graph \((\mathcal {SG})\). Further, the threshold dimension and threshold partition number of \(\mathcal {CPFG}_{s}\) is defined. Some basic results on threshold dimension and threshold partition number also have been discussed. Finally, an application is presented on this developed concept. Due to the wide range of complex Pythagorean fuzzy sets \((\mathcal {CPFS}_{s})\), it is obvious that \(\mathcal {CPFTG}_{s}\) are more helpful and beneficial in modeling a problem as compared to complex fuzzy threshold graphs \((\mathcal {CFTG}_{s})\) and complex intuitionistic fuzzy threshold graphs \((\mathcal {CIFTG}_{s})\).
... However, for any positive integers m and n, the Brualdi-Hoffman problem of graphs of order n and size m is still open. In order to determine the extremal graph in the Brualdi-Hoffman problem under the constraint of order, many reports about the analysis of the extremal graph were presented (see, for example, [1,2,7,22,24,25]). Therefore, it is interesting to investigate the Brualdi-Hoffman problem under additional constraints. ...
Brualdi and Hoffman (1985) proposed the problem of determining the maximal spectral radius of graphs with given size. In this paper, we consider the Brualdi-Hoffman type problem of graphs with given matching number. The maximal $Q$-spectral radius of graphs with given size and matching number is obtained, and the corresponding extremal graphs are also determined.
... For example, Bipartite threshold graphs were studied in [19]. Andelic and Simic [9] presented properties of threshold graphs. Especially, Samanta and Pal [45] first introduced the fuzzy threshold graph in 2011. ...
... As it was observed in [10] (see also [2,7]), the vertices of any connected threshold graph G can be partitioned into h non-empty coduplication classes V 1 , . . . , V h and h non-empty duplication classes U 1 , . . . ...
This paper deals with the eigenvalues of the adjacency matrices of threshold graphs for which $-1$ and $0$ are considered as trivial eigenvalues. We show that among threshold graphs on a fixed number of vertices, the unique connected anti-regular graph has the smallest positive eigenvalue and the largest non-trivial negative eigenvalue. It follows that threshold graphs have no non-trivial eigenvalues in the interval $\left[(-1-\sqrt{2})/2,\,(-1+\sqrt{2})/2\right]$. These results confirm two conjectures by Aguilar, Lee, Piato, and Schweitzer.
... Note that by Dilworth's theorem, ∇(G) is equal to the maximum size of an antichain of V (G) with respect to the vicinal preorder. (Structure of threshold graphs) As it was observed in [20] (see also [1,13]), the vertices of any threshold graph G can be partitioned into t non-empty coduplication classes V 1 , . . . , V t and t non-empty duplication classes U 1 , . . . ...
A cograph is a simple graph which contains no path on 4 vertices as an induced subgraph. The vicinal preorder on the vertex set of a graph is defined in terms of inclusions among the neighborhoods of vertices. The minimum number of chains with respect to the vicinal preorder required to cover the vertex set of a graph $G$ is called the Dilworth number of $G$. We prove that for any cograph $G$, the multiplicity of any eigenvalue $\lambda\ne0,-1$, does not exceed the Dilworth number of $G$ and show that this bound is best possible. We give a new characterization of the family of cographs based on the existence of a basis consisting of weight 2 vectors for the eigenspace of either $0$ or $-1$ eigenvalues. This partially answers a question of G. F. Royle (2003).
... Let G be a graph of order n and G ′ be an induced subgraph of G of order n ′ . If λ 1 ≥ λ 2 ≥ · · · ≥ λ n and λ ′ 1 ≥ λ ′ 2 ≥ · · · ≥ λ ′ n ′ are their eigenvalues respectively, then (2) λ i ≥ λ ′ i ≥ λ n−n ′ +i for i = 1, 2, . . . , n ′ . ...
A graph is called a chain graph if it is bipartite and the neighborhoods of the vertices in each color class form a chain with respect to inclusion. A threshold graph can be obtained from a chain graph by making adjacent all pairs of vertices in one color class. Given a graph $G$, let $\lambda$ be an eigenvalue (of the adjacency matrix) of $G$ with multiplicity $k \geq 1$. A vertex $v$ of $G$ is a downer, or neutral, or Parter depending whether the multiplicity of $\lambda$ in $G-v$ is $k-1$, or $k$, or $k+1$, respectively. We consider vertex types in the above sense in threshold and chain graphs. In particular, we show that chain graphs can have neutral vertices, disproving a conjecture by Alazemi {\em et al.}
... Andelic and Simic [2] studied properties of threshold graphs. Bhutani et al. [3] studied degrees of end nodes and cut nodes in fuzzy graphs. ...
Interval-valued fuzzy threshold graph is an extension of fuzzy threshold graph. In this paper, interval-valued fuzzy threshold graph, interval-valued fuzzy alternating 4-cycle, and interval-valued fuzzy threshold dimension are defined, and certain properties are studied. It is demonstrated that every interval-valued fuzzy threshold graph can be treated as an interval-valued fuzzy split graph.
... First, note that a 1 = d 1 /μ 1 (by applying the eigenvalue equation for any vertex in U 1 ). Applying Lemma 3.1(1) and the inequality κ ≤ ν − 1 +d (conjectured in [4] and first proved in [9] and independently in [2]) we first obtain ...
The Q-index of a simple graph is the largest eigenvalue of its signless Laplacian, or Q-matrix. In our previous paper [the authors, Discrete Appl. Math. 160, No. 4–5, 448–459 (2012; Zbl 1239.05115)] we gave three lower and three upper bounds for the Q-index of nested split graphs, also known as threshold graphs. In this paper, we give another two upper bounds, which are expressed as solutions of cubic equations (in contrast to quadratics from [loc cit.]). Some computational results are also included.
... [4]); here ∆ denotes the maximal vertex degree of a graph in question. The following bound was conjectured in [3], and later proved in [6] (see also [2]): κ ≤ ν − 1 +d, (1.2) whered is the average (vertex) degree of a graph. Many other bounds on Q -index for arbitrary graphs (usually connected ones) can be found in [5]. ...
The Q-index of a simple graph G is the largest eigenvalue of the matrix Q, the signless Laplacian of G. It is well-known that in the set of connected graphs with fixed order and size, the graphs with maximal Q-index are the nested split graphs (also known as threshold graphs). In this paper we focus our attention on the eigenvector techniques for getting some (lower and upper) bounds on the Q-index of nested split graphs. In addition, we give some computational results in order to compare these bounds.
The prime objective of this paper is to introduce a new model of threshold graphs in the complex fuzzy environment that has the capability of handling two-dimensional vague data. The proposed model is called as complex fuzzy threshold graph (CFTG). We introduce certain concepts related to CFTGs including, complex fuzzy alternating 4-cycle, complex fuzzy threshold dimension and complex fuzzy partition number of complex fuzzy graphs. We illustrate these concepts with examples. Further, we investigate some of their properties and discuss proficiency of the presented approach by investigating potential applications related to water source controlling system and internet service providers.
Brualdi and Hoffman [On the spectral radius of (0, 1)-matrices. Linear Algebra Appl. 1985;65:133–146] proposed the problem of determining the maximal spectral radius of graphs with a given size. In this paper, we consider the Brualdi–Hoffman-type problem for graphs with a given matching number. The maximal Q-spectral radius of graphs with a given size and matching number is obtained, and the corresponding extremal graphs are completely determined.
In 1973, Chvatal and Hammer [2] introduced threshold graphs and applied them in set packing problems. This particular class of graphs has many applications in several applied areas such as computer science, scheduling theory, psychology, etc. These graphs are also used in control theory, particularly to control the flow of information between processors, traffic lights controlling system to manage the flow of the traffic, etc. Ordman [4] established the use of these graphs in resource allocation problems. In this chapter, fuzzy threshold graph (FTG) is defined and some of its important properties are investigated. Here, we use the notation \(\mathscr {C}_n\), \(\mathscr {P}_n\), \(\mathscr {K}_n\) for fuzzy cycle, fuzzy path, complete fuzzy graph respectively with n vertices. This work is mainly from [8, 9]. Related works are available in [1, 3, 6, 10, 11].
This book provides an extensive set of tools for applying fuzzy mathematics and graph theory to real-life problems. Balancing the basics and latest developments in fuzzy graph theory, this book starts with existing fundamental theories such as connectivity, isomorphism, products of fuzzy graphs, and different types of paths and arcs in fuzzy graphs to focus on advanced concepts such as planarity in fuzzy graphs, fuzzy competition graphs, fuzzy threshold graphs, fuzzy tolerance graphs, fuzzy trees, coloring in fuzzy graphs, bipolar fuzzy graphs, intuitionistic fuzzy graphs, m-polar fuzzy graphs, applications of fuzzy graphs, and more. Each chapter includes a number of key representative applications of the discussed concept. An authoritative, self-contained, and inspiring read on the theory and modern applications of fuzzy graphs, this book is of value to advanced undergraduate and graduate students of mathematics, engineering, and computer science, as well as researchers interested in new developments in fuzzy logic and applied mathematics.
This paper deals with the eigenvalues of the adjacency matrices of threshold graphs for which −1 and 0 are considered as trivial eigenvalues. We show that threshold graphs have no non-trivial eigenvalues in the interval [(−1−2)/2,(−1+2)/2]. This confirms a conjecture by Aguilar, Lee, Piato, and Schweitzer (2018).
A graph is called a chain graph if it is bipartite and the neighbourhoods of the vertices in each colour class form a chain with respect to inclusion. A threshold graph can be obtained from a chain graph by making adjacent all pairs of vertices in one colour class. Given a graph G, let λ be an eigenvalue (of the adjacency matrix) of G with multiplicity k≥1. A vertex v of G is a downer, or neutral, or Parter depending whether the multiplicity of λ in G−v is k−1, or k, or k+1, respectively. We consider vertex types in the above sense in threshold and chain graphs. In particular, we show that chain graphs can have neutral vertices, disproving a conjecture by Alazemi et al.
This paper deals with the well known classes of threshold and difference graphs, both characterized by separators, i.e. node weight functions and thresholds. We design an efficient algorithm to find the minimum separator, and we show how to maintain minimum its value when the input (threshold or difference) graph is fully dynamic, i.e. edges/nodes are inserted/removed. Moreover, exploiting the data structure used for maintaining the minimality of the separator, we study the disjoint union and the join of two threshold graphs, showing that the resulting graphs are threshold signed graphs, i.e. a superclass of both threshold and difference graphs. Finally, we consider the complement operation on all the three introduced classes of graphs.
All these operations produce in output the modified graph in terms of their separator and require time linear w.r.t. the number of different degrees. We observe that recomputing from scratch the separator would run either in linear (for threshold and difference graphs) or quadratic (for threshold signed graphs) time w.r.t. the number of nodes of the graph.
This part of our work further extends our project of building a new spectral theory of graphs (based on the signless Laplacian) by some results on graph angles, by several comments and by a short survey of recent results.
A spectral graph theory is a theory in which graphs are studied by means of eigenvalues of a matrix M which is in a prescribed way defined for any graph. This theory is called M-theory. We outline a spectral theory of graphs based on the signless Laplacians Q and compare it with other spectral theories, in particular to those based on the adjacency matrix A and the Laplacian L. As demonstrated in the first part, the Q-theory can be constructed in part using various connections to other theories: equivalency with A-theory and L-theory for regular graphs, common features with L-theory for bipartite graphs, general analogies with A-theory and analogies with A-theory via line graphs and subdivision graphs. In this part, we introduce notions of enriched and restricted spectral theories and present results on integral graphs, enumeration of spanning trees, characterizations by eigenvalues, cospectral graphs and graph angles.
We study the signless Laplacian spectral radius of graphs and prove three conjectures of Cvetković, Rowlinson, and Simić [Eigenvalue bounds for the signless Laplacian, Publ. Inst. Math., Nouv. Sér. 81(95) (2007), 11–27].
The largest eigenvalue, or index, of simple graphs is extensively studied in literature. Usually, the authors consider the graphs from some fixed class and identify within it those graphs with maximal (or minimal) index. So far maximal graphs with fixed order, or with fixed size, are identified, but not maximal connected graphs with fixed order and size. In this paper we add some new observations related to the structure of the latter graphs.
We determine the maximum spectral radius for (0,1)-matrices with k2 andk2+1 1's, respectively, and for symmetric (0,1)-matrices with zero trace and 1's (graphs with e edges). In all cases, equality is characterized.
We extend our previous survey of properties of spectra of signless Laplacians of graphs. Some new bounds for eigenvalues are given, and the main result concerns the graphs whose largest eigenvalue is maximal among the graphs with fixed numbers of vertices and edges. The results are presented in the context of a number of computer-generated conjectures.
Given an m × n zero-one matrix A we ask whether there is a single linear inequality ax ≤b whose zero-one solutions are precisely the zero-one solutions of Ax ≤e. We develop an algorithm for answering this question in 0(mn 2) steps and investigate other related problems. Our results may be interpreted in terms of graph theory and threshold logic.
We survey properties of spectra of signless Laplacians of graphs and discuss possibilities for developing a spectral theory of graphs based on this matrix. For regular graphs the whole existing theory of spectra of the adjacency matrix and of the Laplacian matrix transfers directly to the signless Laplacian, and so we consider arbitrary graphs with special emphasis on the non-regular case. The results which we survey (old and new) are of two types: (a) results obtained by applying to the signless Laplacian the same reasoning as for corresponding results concerning the adjacency matrix, (b) results obtained indirectly via line graphs. Among other things, we present eigenvalue bounds for several graph invariants, an interpretation of the coefficients of the characteristic polynomial, a theorem on powers of the signless Laplacian and some remarks on star complements.
We consider four conjectures related to the largest eigenvalue of (the adjacency matrix of) a graph (i.e., to the index of the graph). Three of them have been formulated after some experiments with the programming system AutoGraphiX, designed for finding extremal graphs with respect to given properties by the use of variable neighborhood search. The conjectures are related to the maximal value of the irregularity and spectral spread in n-vertex graphs, to a Nordhaus–Gaddum type upper bound for the index, and to the maximal value of the index for graphs with given numbers of vertices and edges. None of the conjectures has been resolved so far. We present partial results and provide some indications that the conjectures are very hard.
In this paper we improve some classical bounds on the greatest eigenvalue of the adjuacency matrix of a graph. We also give inequalities between the eigenvalues and some other parameters. These results allow us to prove some conjectures of the program Graffiti written by Fajtlowicz. Moreover, the study of the spectrum of graphs obtained by some simple constructions yields infinite families of counterexamples for other conjectures of this program.
Let G=(V,E) be a simple graph with n vertices, e edges, and vertex degrees d1,d2,…,dn. Also, let d1, dn be, respectively, the highest degree and the lowest degree of G and mi be the average of the degrees of the vertices adjacent to vertex vi∈V. It is proved thatmax{dj+mj:vj∈V}⩽2en−1+n−2with equality if and only if G is an Sn graph (K1,n−1⊆Sn⊆Kn) or a complete graph of order n−1 with one isolated vertex. Using the above result we establish the following upper bound for the sum of the squares of the degrees of a graph G:∑i=1ndi2⩽e2en−1+n−2n−1d1+(d1−dn)1−d1n−1with equality if and only if G is a star graph or a regular graph or a complete graph Kd1+1 with n−d1−1 isolated vertices. A comparison is made to another upper bound on ∑i=1ndi2, due to de Caen (Discrete Math. 185 (1998) 245). We also present several upper bounds for ∑i=1ndi2 and determine the extremal graphs which achieve the bounds.
Algorithmic Graph Theory and Perfect Graphs, second edition
- M C Golumbic
M.C. Golumbic, Algorithmic Graph Theory and Perfect Graphs, second edition, in: Annals of Discrete Mathematics, vol. 57, Elsevier Science B.V.,
Amsterdam, 2004.