Non-regular Graphs Research Articles

Construction of Simultaneous Cospectral Graphs for Adjacency, Laplacian and Normalized Laplacian Matrices

In this paper we construct several classes of non-regular graphs which are co-spectral with respect to all the three matrices, namely, adjacency, Laplacian and normalized Laplacian, and hence we answer a question asked by Butler [?]. We make these constructions starting with two pairs (G1, H1) and (G2, H2) of A-cospectral regular graphs, then considering the subdivision graphs S(Gi) and R-graphs ℛ(Hi), i = 1, 2, and finally making some kind of partial joins between S(G1) and ℛ(G2) and S(H1) and ℛ(H2). Moreover, we determine the number of spanning trees and the Kirchhoff index of the newly constructed graphs.

On the Aα-spectra of some corona graphs

Let [Formula: see text] be a simple, connected graph. The matrix [Formula: see text] is defined as [Formula: see text], [Formula: see text], where [Formula: see text] is the adjacency matrix and [Formula: see text] is the degree matrix of [Formula: see text]. Let [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] denote the neighborhood corona, subdivision vertex corona, subdivision edge corona, central vertex corona and central edge corona of two graphs [Formula: see text] and [Formula: see text], respectively. In this paper, we determine the [Formula: see text]-characteristic polynomial and [Formula: see text]-spectra of non-regular graphs obtained from the above operations. As an application, we construct infinitely many pairs of non-regular [Formula: see text]-cospectral graphs. Also, we estimate the [Formula: see text]-energy of the graphs obtained by operating a regular graph [Formula: see text] and a non-regular graph, [Formula: see text].

Faster quantum sampling of Markov chains in nonregular graphs with fewer qubits

Sampling from the stationary distribution is one of the fundamental tasks of Markov chain-based algorithms and has important applications in machine learning, combinatorial optimization and network science. For the quantum case, qsampling from Markov chains can be constructed as preparing quantum states with amplitudes arbitrarily close to the square root of a stationary distribution instead of classical sampling from a stationary distribution. In this paper, a new qsampling algorithm for all reversible Markov chains is constructed by discrete-time quantum walks and works without any limit compared with existing results. In detail, we build a qsampling algorithm that not only accelerates non-regular graphs but also keeps the speed-up of existing quantum algorithms for regular graphs. In non-regular graphs, the invocation of the quantum fast-forward algorithm accelerates existing state-of-the-art qsampling algorithms for both discrete-time and continuous-time cases, especially on sparse graphs. Compared to existing algorithms we reduce log n, where n is the number of graph vertices. In regular graphs, our result matches other quantum algorithms, and our reliance on the gap of Markov chains achieves quadratic speedup compared with classical cases. For both cases, we reduce the number of ancilla qubits required compared to the existing results. In some widely used graphs and a series of sparse graphs where stationary distributions are difficult to reach quickly, our algorithm is the first algorithm to achieve complete quadratic acceleration (without log factor) over the classical case without any limit. To enlarge success probability amplitude amplification is introduced. We construct a new reflection on stationary state with fewer ancilla qubits and think it may have independent application.

The [formula omitted]-spectral radius of nonregular graphs (digraphs) and maximum degree (outdegree)

For 0≤α<1, let Aα(G)=αD(G)+(1−α)A(G) be the Aα-matrix of G, where D(G) is the diagonal degree matrix of G and A(G) is the adjacency matrix of G Let λα(G) denote the Aα-spectral radius of a graph G. Let G be a k-connected nonregular graph with n vertices, m edges, maximum degree Δ and minimum degree δ. In this paper, we show that Δ−λα(G)>(1−α)(nΔ−2m)k2(nΔ−2m)(n−1)2−(Δ−k+1)(n−k−1)+(1−α)nk2,which extends the result on the spectral radius of Xue and Liu (0000) and improves the result on the signless Laplacian spectral radius of Shiu et al. (2017). Furthermore, we also prove that Δ−λα(G)>(1−α)(nΔ−2m)k2(nΔ−2m)(n−Δ+2k−2)(n−δ−1)+k2+(1−α)nk2,which extends the result on the signless Laplacian spectral radius of Ning et al. (2018).Let Γ be a k-strong nonregular digraph of order n, size m, and maximum outdegree Δ+. For 0≤α<1, let Aα(Γ)=αD(Γ)+(1−α)A(Γ) be the Aα-matrix of Γ, where D(Γ) is the diagonal matrix of its vertex outdegrees and A(Γ) is the adjacency matrix of Γ. Denote by λα(Γ) the Aα-spectral radius of a digraph Γ. We prove that Δ+−λα(Γ)>(1−α)(nΔ+−m)k22(nΔ+−m)(n−1)2−(Δ+−k+1)(n−k−1)+(1−α)nk2,which improves the result of Xi and Wang (2020). In the end of the paper, related problem is mentioned.

The replicator equation in stochastic spatial evolutionary games

We study the multi-type stochastic evolutionary game with death–birth updating in expanding spatial populations of size N→∞. The model is a voter model perturbation. For typical eligible populations, we require perturbation strengths satisfying 1/N≪w≪1. Under these conditions, the main results prove that the vector density processes of type obey an extended replicator equation in the limit, and the normalized fluctuations converge to a Gaussian process subject to the Wright–Fisher covariance function. In particular, we obtain a positive resolution of a conjecture from [34] that the replicator equation extends to many non-regular graphs.

On Computing Techniques for Sombor Index of Some Graphs

In all types of topological indicators, degree-based indicators play a major role in chemical graph theory. The topological index is a fixed numeric value associated with graph isomerism. Firstly, in 1972, the concept of degree-based index was developed by Gutman and Trinajstic. These degree-based indices are divided into two ways, namely, degree and connection number. These degree-based graph indices are positive-valued for non-regular graphs and zero for regular graphs. In this article, we discussed the degree-based Sombor, reduced Sombor, and average Sombor indices for wheel graph, gear graph, helm graph, flower graph, sunflower graph, and lobster graph.

Flexible list colorings in graphs with special degeneracy conditions

AbstractFor a given , we say that a graph is ‐flexibly ‐choosable if the following holds: for any assignment of color lists of size on , if a preferred color from a list is requested at any set of vertices, then at least of these requests are satisfied by some ‐coloring. We consider the question of flexible choosability in several graph classes with certain degeneracy conditions. We characterize the graphs of maximum degree that are ‐flexibly ‐choosable for some , which answers a question of Dvořák Norin, and Postle [List coloring with requests, JGT 2019]. In particular, we show that for any , any graph of maximum degree that is not isomorphic to is ‐flexibly ‐choosable. Our fraction of is within a constant factor of being the best possible. We also show that graphs of treewidth 2 are ‐flexibly 3‐choosable, answering a question of Choi et al. [Flexibility of planar graphs‐sharpening the tools to get lists of size four, DAM 2022], and we give conditions for list assignments by which graphs of treewidth are ‐flexibly ‐choosable. We show furthermore that graphs of treedepth are ‐flexibly ‐choosable. Finally, we introduce a notion of flexible degeneracy, which strengthens flexible choosability, and we show that apart from a well‐understood class of exceptions, 3‐connected nonregular graphs of maximum degree are flexibly ‐degenerate.

Robust Connectivity of Graphs on Surfaces

Let $\Lambda(T)$ denote the set of leaves in a tree $T$. One natural problem is to look for a spanning tree $T$ of a given graph $G$ such that $\Lambda(T)$ is as large as possible. This problem is called maximum leaf number, and it is a well-known NP-hard problem. Throughout recent decades, this problem has received considerable attention, ranging from pure graph theoretic questions to practical problems related to the construction of wireless networks. Recently, a similar but stronger notion was defined by Bradshaw, Masa\v{r}\'ik, and Stacho [Flexible List Colorings in Graphs with Special Degeneracy Conditions, ISAAC 2020]. They introduced a new invariant for a graph $G$, called the robust connectivity and written $\kappa_\rho(G)$, defined as the minimum value $\frac{|R \cap \Lambda (T)|}{|R|}$ taken over all nonempty subsets $R\subseteq V(G)$, where $T = T(R)$ is a spanning tree on $G$ chosen to maximize $|R \cap \Lambda(T)|$. Large robust connectivity was originally used to show flexible choosability in non-regular graphs. In this paper, we investigate some interesting properties of robust connectivity for graphs embedded in surfaces. We prove a tight asymptotic bound of $\Omega(\gamma^{-\frac{1}{r}})$ for the robust connectivity of $r$-connected graphs of Euler genus $\gamma$. Moreover, we give a surprising connection between the robust connectivity of graphs with an edge-maximal embedding in a surface and the surface connectivity of that surface, which describes to what extent large induced subgraphs of embedded graphs can be cut out from the surface without splitting the surface into multiple parts. For planar graphs, this connection provides an equivalent formulation of a long-standing conjecture of Albertson and Berman [A conjecture on planar graphs, 1979], which states that every planar graph on $n$ vertices contains an induced forest of size at least $n/2$.

Regular graphs with equal matching number and independence number

Let r≥3 be an integer and G be a graph. Let δ(G),Δ(G), α(G), and μ(G) denote the minimum degree, maximum degree, independence number, and matching number of G, respectively. Recently, Caro, Davila and Pepper proved δ(G)α(G)≤Δ(G)μ(G). Mohr and Rautenbach characterized the extremal graphs within the classes of all non-regular graphs and 3-regular graphs. In this note, we characterize the extremal graphs within the classes of all r-regular graphs in terms of the Gallai–Edmonds Structure Theorem, which extends Mohr and Rautenbach’s result.

Construction And Spectra Of Non-Regular Minimal Graphs

The number of distinct eigenvalues of the adjacency matrix of graph G is bounded below by d(G)+1, where d is the diameter of the graph. Graphs attaining this lower bound are known as minimal graphs. The spectrum of graph G, where G is a simple and undirected graph is the collection of different eigenvalues of the adjacency matrix with their multiplicities. This paper deals with the construction of non-regular minimal graphs, together with the study of their characteristic polynomial and spectra.

On Normalized Laplacian, Degree-Kirchhoff Index of the Strong Prism of Generalized Phenylenes*

The eigenvalues of normalized Laplacian are extremely important for analyzing the structure properties of non-regular graphs. A molecular graph of generalized phenylene Ln is a linear chain containing n hexagons and n quadrangles. The strong prism of a graph G is the strong product of the complete graph of order 2 and G. Let be the strong prism of generalized phenylene Ln . In this paper, the normalized Laplacian spectra of were scrutinized by using decomposition theorem for the corresponding matrices. Further, the explicit formulae of the degree-Kirchhoff index and the number of spanning trees for were obtained.

On Normalized Laplacians, Degree-Kirchhoff Index and Spanning Tree of Generalized Phenylene

The normalized Laplacian is extremely important for analyzing the structural properties of non-regular graphs. The molecular graph of generalized phenylene consists of n hexagons and 2n squares, denoted by Ln6,4,4. In this paper, by using the normalized Laplacian polynomial decomposition theorem, we have investigated the normalized Laplacian spectrum of Ln6,4,4 consisting of the eigenvalues of symmetric tri-diagonal matrices LA and LS of order 4n+1. As an application, the significant formula is obtained to calculate the multiplicative degree-Kirchhoff index and the number of spanning trees of generalized phenylene network based on the relationships between the coefficients and roots.

Quantum Markov Chains on Comb Graphs: Ising Model

In the present paper, we construct quantum Markov chains (QMC) over the Comb graphs. As an application of this construction, it is proved the existence of the disordered phase for the Ising type models (within QMC scheme) over the Comb graphs. Moreover, it is also established that the associated QMC has clustering property with respect to translations of the graph. We stress that this paper is the first one where a nontrivial example of QMC over non-regular graphs is given.

On Ve-Degree-Based Irregularity Properties of the Crystallographic Structure of Molecules

Irregularity indices are usually used for quantitative characterization of the topological structure of nonregular graphs. In numerous applications and problems in material engineering and chemistry, it is useful to be aware that how irregular a molecular structure is? In this paper, we are interested in formulating closed forms of irregularity measures of some of the crystallographic structures of Cu 2 O p , q , r and crystallographic structure of titanium difluoride of T i F 2 p , q , r . These theoretical conclusions provide practical guiding significance for pharmaceutical engineering and complex network and quantify the degree of folding of long organic molecules.

A characterization of graphs with given maximum degree and smallest possible matching number

In 2018 Henning and Yeo (2018) [9] proved that if G is a connected graph of order n, size m and maximum degree k where k≥3 is odd, then α′(G)≥(k−1k(k2−3))n+(k2−k−2k(k2−3))m−k−1k(k2−3), where α′(G) is the matching number of G. For k≥3 odd, the k-regular graphs achieving equality in this bound were characterized by O and West (2010) [12]. In this paper we extend this result due to O and West. For k≥3 odd, we characterize all non-regular graphs with maximum degree k that achieve equality in the Henning-Yeo lower bound on the matching number. Together with the O and West characterization of the regular graphs, this provides a complete characterization.

Distance Based Topological Indices And Regular Graphs

In this article, we are using the regular graph of even number of vertices and computing the distance balanced graphs. First we take a graph for satisfying regular definition and then we compute the Mostar index of that particular graph. If the Mostar index of that particular graph is zero, then the graph is said to be a distance balanced graph. So we discuss first distance balanced graph. Suppose if we delete one edge in that particular graph, that is non-regular graph, we can verify the balanced graph is whether distance balanced graph or not. We discuss and compute the Mostar index of certain regular and non-regular graphs are balanced distance or not. Finally we see few theorems are related in this topic. So in this paper, we study some distance based topological indices for regular graphs and also cubic graphs.

Irregularity Measures of Subdivision Vertex-Edge Join of Graphs

The study of graphs and networks accomplished by topological measures plays an applicable task to obtain their hidden topologies. This procedure has been greatly used in cheminformatics, bioinformatics, and biomedicine, where estimations based on graph invariants have been made available for effectively communicating with the different challenging tasks. Irregularity measures are mostly used for the characterization of the nonregular graphs. In several applications and problems in various areas of research like material engineering and chemistry, it is helpful to be well-informed about the irregularity of the underline structure. Furthermore, the irregularity indices of graphs are not only suitable for quantitative structure-activity relationship (QSAR) and quantitative structure-property relationship (QSPR) studies but also for a number of chemical and physical properties, including toxicity, enthalpy of vaporization, resistance, boiling and melting points, and entropy. In this article, we compute the irregularity measures including the variance of vertex degrees, the total irregularity index, the σ irregularity index, and the Gini index of a new graph operation.

Bounding and approximating minimum maximal matchings in regular graphs

The edge domination number γe(G) of a graph G is the minimum size of a maximal matching in G. It is well known that this parameter is computationally very hard, and several approximation algorithms and heuristics have been studied. In the present paper, we provide best possible upper bounds on γe(G) for regular and non-regular graphs G in terms of their order and maximum degree. Furthermore, we discuss algorithmic consequences of our results and their constructive proofs.

Irregularities of TUC4 and TUAC6 nanotubes

A graph is said to be regular if all its vertices have the same degree, otherwise, it is irregular. Irregularity indices are usually used for quantitative characterization of the topological structure of nonregular graphs. In numerous applications and problems in material engineering and chemistry, it is useful to be aware that how irregular a molecular structure is. Furthermore, evaluations of the irregularity of underline molecular graphs could be valuable for QSAR/QSPR studies, and for the expressive determines of chemical and physical properties, such as enthalpy of vaporization, toxicity, resistance, entropy, melting, and boiling points. In this paper, we find some irregularity measures of nanotubes. We compute 16 irregularities of TUC4 and TUAC6 nanotubes. We also compare the computed irregularities by plotting our results using software Mathematica.

Two irregularity measures possessing high discriminatory ability

An $n$-vertex graph whose degree set consists of exactly $n-1$ elements is called antiregular graph. Such type of graphs are usually considered opposite to the regular graphs. An irregularity measure ($IM$) of a connected graph $G$ is a non-negative graph invariant satisfying the property: $IM(G) = 0$ if and only if $G$ is regular. The total irregularity of a graph $G$, denoted by $irr_t(G)$, is defined as $irr_t(G)= \sum_{\{u,v\} \subseteq V(G)} |d_u - d_v|$ where $V(G)$ is the vertex set of $G$ and $d_u$, $d_v$ denote the degrees of the vertices $u$, $v$, respectively. Antiregular graphs are the most nonregular graphs according to the irregularity measure $irr_t$; however, various non-antiregular graphs are also the most nonregular graphs with respect to this irregularity measure. In this note, two new irregularity measures having high discriminatory ability are devised. Only antiregular graphs are the most nonregular graphs according to the proposed measures.