Connected Threshold Graphs Research Articles

Combinatorial versus algebraic formulae for the Moore–Penrose inverse of a Laplacian matrix of a threshold graph

We present four different approaches to calculate the Moore–Penrose inverse of a Laplacian matrix of a connected threshold graph. The first one is combinatorial and it benefits from the combinatorial structure of a threshold graph, while the second three are of algebraic nature and explore the particular structure of a matrix in question.

Realization of zero-divisor graphs of finite commutative rings as threshold graphs

Let R be a finite commutative ring with unity, and let $$G = (V, E)$$ be a simple graph. The zero-divisor graph, denoted by $$\Gamma (R)$$ is a simple graph with vertex set as R, and two vertices $$x, y \in R$$ are adjacent in $$\Gamma (R)$$ if and only if $$xy=0$$ . In [5], the authors have studied the Laplacian eigenvalues of the graph $$\Gamma (\mathbb {Z}_{n})$$ and for distinct proper divisors $$d_1, d_2, \cdots , d_k$$ of n, they defined the sets as, $$\mathcal {A}_{d_i} = \{x \in \mathbb {Z}_{n} : (x, n) = d_i\}$$ , where (x, n) denotes the greatest common divisor of x and n. In this paper, we show that the sets $$\mathcal {A}_{d_i}$$ , $$1 \le i \le k$$ are actually orbits of the group action: $$Aut(\Gamma (R)) \times R \longrightarrow R$$ , where $$Aut(\Gamma (R))$$ denotes the automorphism group of $$\Gamma (R)$$ . Our main objective is to determine new classes of threshold graphs, since these graphs play an important role in several applied areas. For a reduced ring R, we prove that $$\Gamma (R)$$ is a connected threshold graph if and only if $$R\cong {F}_{q}$$ or $$R\cong {F}_2 \times {F}_{q}$$ . We provide classes of threshold graphs realized by some classes of local rings. Finally, we characterize all finite commutative rings with unity of which zero-divisor graphs are not threshold.

On the minimum number of distinct eigenvalues of a threshold graph

For a graph G, we associate a family of real symmetric matrices, S(G), where for any A∈S(G), the location of the nonzero off-diagonal entries of A are governed by the adjacency structure of G. Let q(G) be the minimum number of distinct eigenvalues over all matrices in S(G). In this work, we give a characterization of all connected threshold graphs G with q(G)=2. Moreover, we study the values of q(G) for connected threshold graphs with trace 2, 3, n−2, n−3, where n is the order of threshold graph. The values of q(G) are determined for all connected threshold graphs with 7 and 8 vertices with two exceptions. Finally, a sharp upper bound for q(G) over all connected threshold graph G is given.

Extremal graphs for the Tutte polynomial

A graph transformation called the compression of a graph G is known to decrease the number of spanning trees, the all-terminal reliability, and the magnitude of the coefficients of the chromatic polynomial of a graph G. All of these graph parameters can be derived from the Tutte polynomial of G, and in this paper we determine more generally compression's effect on the Tutte polynomial, recovering the previous results and obtaining similar results for a wide variety of other graph parameters derived from the Tutte polynomial. Since any simple connected graph can be transformed into a connected threshold graph via a series of compressions, this gives that threshold graphs are extremal simple graphs for all of the parameters considered.

Algorithmic Aspects of Outer-Independent Total Roman Domination in Graphs

For a simple, undirected graph [Formula: see text], a function [Formula: see text] which satisfies the following conditions is called an outer-independent total Roman dominating function (OITRDF) of [Formula: see text] with weight [Formula: see text]. (C1) For all [Formula: see text] with [Formula: see text] there exists a vertex [Formula: see text] such that [Formula: see text] and [Formula: see text], (C2) The induced subgraph with vertex set [Formula: see text] has no isolated vertices and (C3) The induced subgraph with vertex set [Formula: see text] is independent. For a graph [Formula: see text], the smallest possible weight of an OITRDF of [Formula: see text] which is denoted by [Formula: see text], is known as the outer-independent total Roman domination number of [Formula: see text]. The problem of determining [Formula: see text] of a graph [Formula: see text] is called minimum outer-independent total Roman domination problem (MOITRDP). In this article, we show that the problem of deciding if [Formula: see text] has an OITRDF of weight at most [Formula: see text] for bipartite graphs and split graphs, a subclass of chordal graphs is NP-complete. We also show that MOITRDP is linear time solvable for connected threshold graphs and bounded treewidth graphs. Finally, we show that the domination and outer-independent total Roman domination problems are not equivalent in computational complexity aspects.

Notes on Hamiltonian threshold and chain graphs

<abstract> We revisit results obtained in ^{[<xref ref-type="bibr" rid="b1">1</xref>]}, where several necessary and necessary and sufficient conditions for a connected threshold graph to be Hamiltonian were obtained. We present these results in new forms, now stated in terms of structural parameters that uniquely define the threshold graph and we extend them to chain graphs. We also identify the chain graph with minimum number of Hamilton cycles within the class of Hamiltonian chain graphs of a given order. </abstract>

Minimal Laplacian controllability problems of threshold graphs

This study is concerned with the minimal controllability problem of a connected threshold graph following the Laplacian dynamics. The goal is to find the minimum number of controllers and a small set of vertices for the controllers to connect to render the graph Laplacian controllable. A simple algorithm is provided to generate a spanning set of orthogonal Laplacian eigenvectors of the graph from a straightforward computation on its Laplacian matrix. A necessary and sufficient condition for the graph to be Laplacian controllable is then proposed. The condition suggests that the minimum number of controllers to make a connected threshold graph Laplacian controllable is the maximum multiplicity of Laplacian eigenvalues of the graph, and this minimum can be achieved using a binary control matrix. If a controller can be connected to one vertex only, the minimum number is the difference between the number of vertices in the graph and the number of vertices with different degrees. The condition also implies that the controllers ensuring the Laplacian controllability should be connected to the vertices with repeating degrees to break the symmetry of the network topology. Several examples are provided to illustrate the authors' results.

On the Eigenvalues Distribution in Threshold Graphs

A threshold graph G of order n is defined by binary sequence of length n. In this paper, we consider the adjacent matrix of a connected threshold graph, and give the eigenvalues distribution in threshold graphs. Moreover, we pick out all the threshold graphs with distinct eigenvalues, and determine the HOMO–LUMO index of threshold graphs.

On the distance spectra of threshold graphs

A graph is called a threshold graph if it does not contain induced C4, P4 or 2K2. Such graphs have numerous applications in computer science and psychology, and they also have nice spectral properties. In this paper, we consider the distance matrix of a connected threshold graph. We show that there are no distance eigenvalues of threshold graphs lying in the interval (−2,−1) and all the eigenvalues, other than −2 or −1, are simple. Besides, we determine all threshold graphs with distinct distance eigenvalues.

On the normalized spectrum of threshold graphs

In this article we investigate normalized adjacency eigenvalues (simply normalized eigenvalues) and normalized adjacency energy of connected threshold graphs. A threshold graph can always be represented as a unique binary string. Certain eigenvalues are obtained directly from its binary representation and the rest of the eigenvalues are evaluated from its normalized equitable partition matrix. Finally, we characterize threshold graphs with at most five distinct eigenvalues.

Distribution of Laplacian eigenvalues of graphs

Let G be a graph of order n with m edges and clique number ω. Let μ1≥μ2≥…≥μn=0 be the Laplacian eigenvalues of G and let σ=σ(G)(1≤σ≤n) be the largest positive integer such that μσ≥2mn. In this paper we study the relation between σ and ω. In particular, we provide the answer to Problem 2.3 raised in Pirzada and Ganie (2015) [15]. Moreover, we characterize all connected threshold graphs with σ<ω−1, σ=ω−1 and σ>ω−1. We obtain Nordhaus–Gaddum-type results for σ. Some relations between σ with other graph invariants are obtained.

Extremal Laplacian energy of threshold graphs

Let G be a connected threshold graph of order n with m edges and trace T. In this paper we give a lower bound on Laplacian energy in terms of n, m and T of G. From this we determine the threshold graphs with the first four minimal Laplacian energies. Moreover, we obtain the threshold graphs with the largest and the second largest Laplacian energies.

Threshold graphs of maximal Laplacian energy

The Laplacian energy of a graph is defined as the sum of the absolute values of the differences of average degree and eigenvalues of the Laplacian matrix of the graph. This spectral graph parameter is upper bounded by the energy obtained when replacing the eigenvalues with the conjugate degree sequence of the graph, in which the ith number counts the nodes having degree at least i. Because the sequences of eigenvalues and conjugate degrees coincide for the class of threshold graphs, these are considered likely candidates for maximizing the Laplacian energy over all graphs with given number of nodes. We do not answer this open problem, but within the class of threshold graphs we give an explicit and constructive description of threshold graphs maximizing this spectral graph parameter for a given number of nodes, for given numbers of nodes and edges, and for given numbers of nodes, edges and trace of the conjugate degree sequence in the general as well as in the connected case. In particular this positively answers the conjecture that the pineapple maximizes the Laplacian energy over all connected threshold graphs with given number of nodes.

Eigenvalues and energy in threshold graphs

Assuming a uniform random model of selecting creation sequences, we show that almost every connected threshold graph has more negative than positive eigenvalues. We show that no threshold graphs have eigenvalues in (−1,0). Sequences of equienergetic graphs are given including the striking result that for all n≥3, there exist n−1 threshold graphs of order n2, pairwise noncospectral, each equienergetic to Kn2. For all n>8, we find an n-vertex hyperenergetic threshold graph.

Maximum Laplacian energy among threshold graphs

We impose some restrictions on the spectra of a threshold graph, allowing us to compute its Laplacian energy. We then show that the pineapple with clique number 1+⌊2n3⌋ has largest Laplacian energy among all pineapples with n vertices and among all the graphs satisfying those conditions. We also find large families of threshold graphs that satisfy those restrictions and exhibit pairs of such graphs with the same Laplacian energy and different Laplacian spectra. We show by computation that the above pineapple has maximum Laplacian energy among all connected threshold graphs for all n<35.

On the Spectrum of Threshold Graphs

The antiregular connected graph on r vertices is defined as the connected graph whose vertex degrees take the values of r−1 distinct 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 number r of 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.

Cubicity of threshold graphs

We show that the cubicity of a connected threshold graph is equal to ⌈ log 2 α ⌉ , where α is its independence number.