arXiv subjects

Jia-Bao Liu

Publications and source records attributed to Jia-Bao Liu.

At least 19 recordsLinked to original sources

arXiv2025

Some Relations on Paratopisms and An Intuitive Interpretation on the Parastrophes of a Latin Square

This paper will present some intuitive interpretation of the parastrophe transformations of arbitrary Latin square. With this trick, we can generate the parastrophes of arbitrary Latin square directly from the original one without generating the orthogonal array. The relations of isotopisms and parastrophe transformations in composition will also be shown. It will solve the problem that when F1*I1=I2*F2 how can we obtain I2 and F2 from I1 and F1, where I1 and I2 are isotopisms while F1 and F2 are parastrophe transformations and "{*}" is the composition of transformations. These methods could distinctly simplify the computation on a computer for the issues related to main classes of Latin squares. This will improve the efficiency apparently in computation for some related problems.

math.CO↗

arXiv2022

The statistical analysis for Sombor indices in a random polygonal chain networks

The Sombor indices, a new category of degree-based topological molecular descriptors, have been widely investigated due to their excellent chemical applicability. This paper aims to establish Sombor indices distributions in random polygonal chain networks and to achieve expressions of the expected values and variances. The expected values and variances of the Sombor indices for polyonino, pentachain, polyphenyl, and cyclooctane chains are obtained. Since the end connection of a random chain network follows a binomial distribution, the Sombor indices of any chain network follow the normal distribution when the number of polygons connected by the chain, indicated by n, approaches infinity. Keywords: Degree distribution; Polygonal chains; Expected value; Variance; Sombor indices.

math.CO↗

arXiv2022

Metric dimension, doubly resolving set and strong metric dimension for $(C_n\Box P_k)\Box P_m$

A subset $Q = \{q_1, q_2, ..., q_l\}$ of vertices of a connected graph $G$ is a doubly resolving set of $G$ if for any various vertices $x, y \in V(G)$ we have $r(x|Q)-r(y|Q)\neqλI$, where $λ$ is an integer, and $I$ indicates the unit $l$- vector $(1,..., 1)$. A doubly resolving set of vertices of graph $G$ with the minimum size, is denoted by $ψ(G)$. In this work, we will consider the computational study of some resolving sets with the minimum size for $(C_n\Box P_k)\Box P_m$.

math.CO↗

arXiv2022

The (degree)-Kirchhoff index of linear crossed octagonal-quadrilateral networks

The Kirchhoff index and degree-Kirchhoff index have attracted extensive attentions due to its practical applications in complex networks, physics, and chemistry. In 2019, Liu et al. [Int. J. Quantum Chem. 119 (2019) e25971] derived the formula of the degree-Kirchhoff index of linear octagonal-quadrilateral networks. In the present paper, we consider linear crossed octagonal-quadrilateral networks $Q_n$. Explicit closed-form formulas of the Kirchhoff index, the degree-Kirchhoff index, and the number of spanning trees of $Q_n$ are obtained. Moreover, the Kirchhoff index (resp. degree-Kirchhoff index) of $Q_n$ is shown to be almost 1/4 of its Wiener index (resp. Gutman index).

math.CO↗

arXiv2022

On the Number of Conjugate Classes of Derangements

The number of conjugate classes of derangements of order $n$ is the same as the number $h(n)$ of the restricted partitions with every portion greater than $1$. It is also equal to the number of isotopy classes of $2\times n$ Latin rectangles. Sometimes the exact value is necessary, while sometimes we need the approximation value. In this paper, a recursion formula of $h(n)$ will be obtained, also will some elementary approximation formulae with high accuracy for $h(n)$ be presented. Although we may obtain the value of $h(n)$ in some computer algebra system, it is still meaningful to find an efficient way to calculate the approximate value, especially in engineering, since most people are familiar with neither programming nor CAS software. This paper is mainly for the readers who need a simple and practical formula to obtain the approximate value (without writing a program) with more accuracy, such as to compute the value in an pocket science calculator without programming function. Some methods used here can also be applied to find the fitting functions for some types of data obtained in experiments.

math.CO↗

arXiv2022

The Laplacians, Kirchhoff index and complexity of linear Möbius and cylinder octagonal-quadrilateral networks

Spectrum graph theory not only facilitate comprehensively reflect the topological structure and dynamic characteristics of networks, but also offer significant and noteworthy applications in theoretical chemistry, network science and other fields. Let $L_{n}^{8,4}$ represent a linear octagonal-quadrilateral network, consisting of $n$ eight-member ring and $n$ four-member ring. The Möbius graph $Q_{n}(8,4)$ is constructed by reverse identifying the opposite edges, whereas cylinder graph $Q'_{n}(8,4)$ identifies the opposite edges by order. In this paper, the explicit formulas of Kirchhoff indices and complexity of $Q_{n}(8,4)$ and $Q'_{n}(8,4)$ are demonstrated by Laplacian characteristic polynomials according to decomposition theorem and Vieta's theorem. In surprise, the Kirchhoff index of $Q_{n}(8,4)$($Q'_{n}(8,4)$) is approximately one-third half of its Wiener index as $n\to\infty$.

math.CO↗

arXiv2022

The expected values, variances and limiting distributions of Gutman index, Schultz index, multiplicative degree-Kirchhoff index and additive degree-Kirchhoff index for a class of random chain networks

There has been an upsurge of research on complex networks in recent years. The purpose of this paper is to study the mathematical properties of the random chain networks PGn with the help of graph theory. We first solve the expected value expressions of the Gutman index, Schultz index, multiplicative degree-Kirchhoff index and additive degree-Kirchhoff index, and then we get the explicit expression formulas of their variances. Finally, we find that their limiting distributions all have the probabilistic and statistical significance of normal distribution.

math.CO↗

arXiv2022

The expected values and limiting behaviours for the Gutman index, Schultz index, multiplicative degree-Kirchhoff index and additive degree-kirchhoff index of a random cyclooctane chain

In this paper, we first introduce the explicit analytical formulas for the expected values of the Gutman and Schultz indices for a random cyclooctane chain COCn. Meanwhile, the explicit formulas of the variances of the Gutman and Schultz indices for a random cyclooctane chain are determined and we prove these two indices are asymptotically subject to normal distribution. Furthermore, we are surprised to find the variances of Kf*(COCn) and Kf+(COCn) for a random cyclooctane chain based on the known results of others' paper and they are asymptotically subject to normal distribution.

math.PR↗

arXiv2022

Analyses of Some Structural Properties on a Class of Hierarchical Scale-free Networks

Hierarchical networks actually have many applications in the real world. Firstly, we propose a new class of hierarchical networks with scale-free and fractal structure, which are the networks with triangles compared to traditional hierarchical networks. Secondly, we study the precise results of some structural properties to derive small-world effect and scale-free feature. Thirdly, it is found that the constructed network is sparse through the average degree and density. Fourthly, it is also demonstrated the degree distributions of hub nodes and the bottom nodes are the power law and exponential, respectively. Finally, we prove that clustering coefficient with a definite value z tends to stabilize at a lower bound as t iterates to a certain number, and the average distance of G_{t}^{z} has a increasing relationship along with the value of lnN_{t}.

math.CO↗

arXiv2022

The Normalized Laplacian Spectrum Analysis of Fractal Mobius Octagonal Networks and its Applications

The study and calculation of spectrum of networks can be used to describe networks structure and quantify analysis of networks performance. The fractal Möbius octagonal networks, denoted by $Q_n$, is derived from the inverse identification of the opposite lateral edges of fractal linear octagonal networks. In this paper, the normalized Laplacian spectrum of $Q_n$ is determined by two matrices $\mathcal {L}_A$ and $\mathcal {L}_S$. As an important application of our results, some topological indices (multiplicative degree-Kirchhoff index, the number of spanning trees) formulas of $Q_n$ are obtained.

math.CO↗

arXiv2021

Some resolving parameters in a class of Cayley graphs

Resolving parameters is a fundamental area of combinatorics with applications not only to many branches of combinatorics but also to other sciences. In this article, we construct a class of Toeplitz graphs, and will be denoted by $T_{2n}(W)$, so that they are Cayley graphs. First, we review some of the features of this class of graphs. In fact, this class of graphs are vertex transitive, and by calculating the spectrum of the adjacency matrix related with them, we show that this class of graphs cannot be edge transitive. Moreover, we show that this class of graphs cannot be distance regular, and since the computing resolving parameters of a class of graphs such that are not distance regular is more difficult, then we regard this as justification for our focus on some resolving parameters. In particular, we determine the minimal resolving set, doubly resolving set and strong metric dimension for this class of graphs.

math.CO↗

arXiv2021

Multiset and Mixed Metric Dimension for Starphene and Zigzag-Edge Coronoid

Let $Γ=(V,E)$ be a simple connected graph. A vertex $a$ is said to recognize (resolve) two different elements $b_{1}$ and $b_{2}$ from $V(Γ)\cup E(Γ)$ if $d(a, b_{1})\neq d(a, b_{2}\}$. A subset of distinct ordered vertices $U_{M}\subseteq V(Γ)$ is said to be a mixed metric generator for $Γ$ if each pair of distinct elements from $V\cup E$ are recognized by some element of $U_{M}$. The mixed metric generator with a minimum number of elements is called a mixed metric basis of $Γ$. Then, the cardinality of this mixed metric basis for $Γ$ is called the mixed metric dimension of $Γ$, denoted by $mdim(Γ)$. The concept of studying chemical structures using graph theory terminologies is both appealing and practical. It enables researchers to more precisely and easily examines various chemical topologies and networks. In this paper, we consider two well known chemical structures; starphene $SP_{a,b,c}$ and six-sided hollow coronoid $HC_{a,b,c}$ and respectively compute their multiset dimension and mixed metric dimension.

math.CO↗

arXiv2021

Edge Resolvability for Circular Ladder of Heptagons

A set $\mathbb{Y}$ of elements (vertices or edges) in space is said to be a $generator$ of a metric space if each element of the space is recognized by its distances from the elements of $\mathbb{Y}$, uniquely. The generator with minimum cardinality is known as the $basis$ of the metric space, and this cardinality is the $dimension$ of the given space. In this article, we further discuss these notions with respect to a heptagonal circular ladder. We show that for a heptagonal circular ladder $Γ_{n}$, the edge metric dimension is three and find that it equals its metric dimension. We also introduce a new family of the convex polytope graph (denoted by $Δ_{n}$) from a heptagonal circular ladder and find its metric dimension. Furthermore, we prove that the minimum generator (metric and edge metric) are independent for all of these families of the convex polytopes.

math.CO↗

arXiv2021

Some algebraic properties of a class of integral graphs determined by their spectrum

Let $Γ=(V,E)$ be a graph. If all the eigenvalues of the adjacency matrix of the graph $Γ$ are integers, then we say that $Γ$ is an integral graph. A graph $Γ$ is determined by its spectrum if every graph cospectral to it is in fact isomorphic to it. In this paper, we investigate some algebraic properties of the Cayley graph $Γ=Cay(\mathbb{Z}_{n}, S)$, where $n=p^m$, ($p$ is a prime integer, $m\in\mathbb{N}$) and $S=\{{a}\in\mathbb{Z}_{n}\,|\,\, (a, n)=1\}$. First, we show that $Γ$ is an integral graph. Also we determine the automorphism group of $Γ$. Moreover, we show that $Γ$ and $K_v \bigtriangledownΓ$ are determined by their spectrum.

math.CO↗

arXiv2020

Computing Minimal Doubly Resolving Sets and the Strong Metric Dimension of the Layer Sun Graph and the Line Graph of Layer Sun Graph

Let $G$ be a finite, connected graph of order of at least 2, with vertex set $V(G)$ and edge set $E (G)$. A set $S$ of vertices of the graph $G$ is a doubly resolving set for $G$ if every two distinct vertices of $G$ are doubly resolved by some two vertices of $S$. The minimal doubly resolving set of vertices of graph $G$ is a doubly resolving set with minimum cardinality and is denoted by $ψ(G)$. In this paper, first, we construct a class of graphs of order $2n+ Σ_{r=1}^{k-2}nm^{r}$, denoted by $LSG(n,m, k)$, and call these graphs as the layer Sun graphs with parameters $n$, $m$ and $k$. Moreover, we compute minimal doubly resolving sets and the strong metric dimension of layer Sun graph $LSG(n,m, k)$ and the line graph of the layer Sun graph $LSG(n,m, k)$.

math.CO↗

arXiv2020

The Laplacian spectrum, Kirchhoff index and complexity of the linear heptagonal networks

Let $H_n$ be the linear heptagonal networks with $2n$ heptagons. We study the structure properties and the eigenvalues of the linear heptagonal networks. According to the Laplacian polynomial of $H_n$, we utilize the decomposition theorem. Thus, the Laplacian spectrum of $H_n$ is created by eigenvalues of a pair of matrices: $L_A$ and $L_S$ of order number $5n+1$ and $4n+1$, respectively. On the basis of the roots and coefficients of their characteristic polynomials of $L_A$ and $L_S$, we not only get the explicit forms of Kirchhoff index, but also corresponding total complexity of $H_n$.

math.CO↗

arXiv2020

The Laplacian and normalized Laplacian spectra of Mobius polyomino networks and their applications

Spectral theory has widely used in complex networks and solved some practical problems. In this paper, we investigated the Laplacian and normalized Laplacian spectra of Mobius polyomino networks by using spectral theory. Let Mn denote Mobius polyomino networks (n>=3). As applications of the obtained results, the Kirchhoff index, multiplicative degree-Kirchhoff index, Kemeny's constant and spanning trees of Mn are obtained. Moreover, it is surprising to find that the multiplicative degree-Kirchhoff index of Mn is nine times as much as the Kirchhoff index.

math.SP↗

arXiv2020

Ordering Trees by Their ABC Spectral Radii

Let $G=(V,E)$ be a connected graph, where $V=\{v_1, v_2, \cdots, v_n\}$. Let $d_i$ denote the degree of vertex $v_i$. The ABC matrix of $G$ is defined as $M(G)=(m_{ij})_{n \times n}$, where $m_{ij}=\sqrt{(d_i + d_j -2)/(d_i d_j)}$ if $v_i v_j \in E$, and 0 otherwise. The ABC spectral radius of $G$ is the largest eigenvalue of $M(G)$. In the present paper, we establish two graph perturbations with respect to ABC spectral radius. By applying these perturbations, the trees with the third, fourth, and fifth largest ABC spectral radii are determined.

math.CO↗