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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.

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).

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 M\"{o}bius 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 $n\to\infty$.

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., Comment: arXiv admin note: text overlap with arXiv:2203.12923

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., Comment: 21 pages, 3 figures

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}.

The study and calculation of spectrum of networks can be used to describe networks structure and quantify analysis of networks performance. The fractal M\"{o}bius 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.

The problem of determining resolving sets in graph theory has a long history, as it has many applications in chemistry, robot navigation, combinatorial optimization and utilization of the idea in pattern recognition and processing of images that also provide motivation for founding the theory. Especially, it is well known that these problems are NP hard. In the present work, first we define the structure of the crystal cubic carbon $CCC(n)$ by a new method, and we will find the minimum number of doubly resolving sets and strong resolving sets for the crystal cubic carbon $CCC(n)$. Also, we construct a new class of graphs of order $n+ \Sigma_{p=2}^{k}n^2(n-1)^{p-2}$, is denoted by $LCG(n, k)$ and recall that the layer cycle graph with parameters $n$ and $k$. Moreover, we will compute the minimum number of some resolving sets for the layer cycle graph $LCG(n, k)$.

Let $\Gamma=(V,E)$ be a simple connected graph. A vertex $a$ is said to recognize (resolve) two different elements $b_{1}$ and $b_{2}$ from $V(\Gamma)\cup E(\Gamma)$ if $d(a, b_{1})\neq d(a, b_{2}\}$. A subset of distinct ordered vertices $U_{M}\subseteq V(\Gamma)$ is said to be a mixed metric generator for $\Gamma$ 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 $\Gamma$. Then, the cardinality of this mixed metric basis for $\Gamma$ is called the mixed metric dimension of $\Gamma$, denoted by $mdim(\Gamma)$. 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., Comment: 13 pages, 3 figures, 3 tables

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 $r(x|Q)-r(y|Q)\neq\lambda I$, where $\lambda$ 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 $\psi(G)$. In this work, we will consider the computational study of some resolving sets with the minimum size for $(C_n\Box P_k)\Box P_m$., Comment: arXiv admin note: substantial text overlap with arXiv:2103.15560

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 $\Gamma_{n}$, 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 $\Delta_{n}$) 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., Comment: 3 figures

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 $H_n$.

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.

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.

Recently, Li et al. [Appl. Math. Comput. 382 (2020) 125335] proposed the problem of determining the Kirchhoff index and multiplicative degree-Kirchhoff index of graphs derived from $S_n \times K_2$, the Catersian product of the star $S_n$ and the complete graph $K_2$. In the present paper, we completely solve this problem. That is, the explicit closed-form formulae of Kirchhoff index, multiplicative degree-Kirchhoff index, and number of spanning trees are obtained for some graphs derived from $S_n \times K_2$., Comment: 13 pages, 1 figure

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 $\psi(G)$. In this paper, first, we construct a class of graphs of order $2n+ \Sigma_{r=1}^{k-2}nm^{r}$, 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 $LSG(n,m, k)$.

Let $G=(V,E)$ be a connected graph, where $V=\{v_1, v_2, \cdots, v_n\}$ and $m=|E|$. $d_i$ will denote the degree of vertex $v_i$ of $G$, and $\Delta=\max_{1\leq i \leq n} d_i$. 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 largest eigenvalue of $M(G)$ is called the ABC spectral radius of $G$, denoted by $\rho_{ABC}(G)$. Recently, this graph invariant has attracted some attentions. We prove that $\rho_{ABC}(G) \leq \sqrt{\Delta+(2m-n+1)/\Delta -2}$. As an application, the unique tree with $n \geq 4$ vertices having second largest ABC spectral radius is determined., Comment: 10 pages

The $k$-parallel subdivision graph $S_k(G)$ is generated from $G$ which each edge of $G$ is replaced by $k$ parallel paths of length 2. The $2k$-parallel subdivision graph $S_{2k}(G)$ is constructed from $G$ which each edge of $G$ is replaced by $k$ parallel paths of length 3. In this paper, the normalized Laplacian spectra of $S_k(G)$ and $S_{2k}(G)$ are given. They turn out that the multiplicities of the corresponding eigenvalues are only determined by $k$. As applications, the expected hitting time, the expected commute time and any two-points resistance distance between vertices $i$ and $j$ of $S_k(G)$, the normalized Laplacian spectra of $S_k(G)$ and $S_{2k}(G)$ with $r$ iterations are given. Moreover, the multiplicative degree Kirchhoff index, Kemeny's constant and the number of spanning tress of $S_k(G)$, $S_k^r(G)$, $S_{2k}(G)$ and $S_{2k}^r(G)$ are respectively obtained. Our results have generalized the previous works in Xie et al. and Guo et al. respectively.

Let $\Gamma$ be a simple connected undirected graph with vertex set $V(\Gamma)$ and edge set $E(\Gamma)$. The metric dimension of a graph $\Gamma$ is the least number of vertices in a set with the property that the list of distances from any vertex to those in the set uniquely identifies that vertex. For an ordered subset $W = \{w_1, w_2, ..., w_k\}$ of vertices in a graph $\Gamma$ and a vertex $v$ of $\Gamma$, the metric representation of $v$ with respect to $W$ is the $k$-vector $r(v | W) = (d(v, w_1), d(v, w_2), ..., d(v, w_k ))$. If every pair of distinct vertices of $\Gamma$ have different metric representations then the ordered set $W$ is called a resolving set of $\Gamma$. It is known that the problem of computing this invariant is NP-hard. In this paper, we consider the problem of determining the cardinality $\psi(\Gamma)$ of minimal doubly resolving sets of $\Gamma$, and the strong metric dimension for jellyfish graph $JFG(n, m)$ and cocktail party graph $CP(k+1)$., Comment: arXiv admin note: substantial text overlap with arXiv:1905.10525, arXiv:1908.07854

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., Comment: arXiv admin note: substantial text overlap with arXiv:1905.10527

Let $\Gamma=(V,E)$ be a graph. If all the eigenvalues of the adjacency matrix of the graph $\Gamma$ are integers, then we say that $\Gamma$ is an integral graph. A graph $\Gamma$ 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 $\Gamma=Cay(\mathbb{Z}_{n}, S)$, 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 $\Gamma$ is an integral graph. Also we determine the automorphism group of $\Gamma$. Moreover, we show that $\Gamma$ and $K_v \bigtriangledown\Gamma$ are determined by their spectrum.

Let Ln denote linear octagonal-quadrilateral networks. In this paper, we aim to firstly investigate the Laplacian spectrum on the basis of Laplacian polynomial of Ln. Then, by applying the relationship between the coefficients and roots of the polynomials, the Kirchhoff index and the complexity are determined., Comment: Let Ln denote linear octagonal-quadrilateral networks. In this paper, we aim to firstly investigate the Laplacian spectrum on the basis of Laplacian polynomial of Ln. Then, by applying the relationship between the coefficients and roots of the polynomials, the Kirchhoff index and the complexity are determined

Resistance distance is a novel distance function, also a new intrinsic graph metric, which makes some extensions of ordinary distance. Let On be a linear crossed octagonal graph. Recently, Pan and Li (2018) derived the closed formulas for the Kirchhoff index, multiplicative degree-Kirchhoff index and the number of spanning trees of Hn. They pointed that it is interesting to give the explicit formulas for the Kirchhoff and multiplicative degree-Kirchhoff indices of On. Inspired by these, in this paper, two resistance distance-based graph invariants, namely, Kirchhoff and multiplicative degree-Kirchhoff indices are studied. We firstly determine formulas for the Laplacian (normalized Laplacian, resp.) spectrum of On. Further, the formulas for those two resistance distance-based graph invariants and spanning trees are given. More surprising, we find that the Kirchhoff (multiplicative degree-Kirchhoff, resp.) index is almost one quarter to Wiener (Gutman, resp.) index of a linear crossed octagonal graph., Comment: In this paper, we firstly determine formulas for the Laplacian (normalized Laplacian, resp.) spectrum of On. Further, the formulas for those two resistance distance-based graph invariants and spanning trees are given. More surprising, we find that the Kirchhoff (multiplicative degree-Kirchhoff, resp.) index is almost one quarter to Wiener (Gutman, resp.) index of a linear crossed octagonal graph

Let $A(G)$ be the adjacent matrix and $D(G)$ the diagonal matrix of the degrees of a graph $G$, respectively. For $0 \leq \alpha \leq 1$, the $A_{\alpha}$ matrix $A_{\alpha}(G) = \alpha D(G) +(1-\alpha)A(G)$ is given by Nikiforov. Clearly, $A_{0} (G)$ is the adjacent matrix and $2 A_{\frac{1}{2}}$ is the signless Laplacian matrix. A cactus is a connected graph such that any two of its cycles have at most one common vertex, that is an extension of the tree. The $A_{\alpha}$-spectral radius of a cactus graph with $n$ vertices and $k$ cycles is explored. The outcomes obtained in this paper can imply previous bounds of Nikiforov et al., and Lov\'{a}sz and Pelik\'{a}n. In addition, the corresponding extremal graphs are determined. Furthermore, we proposed all eigenvalues of such extremal cacti. Our results extended and enriched previous known results., Comment: submitted

This paper will present some intuitive interpretation of the adjugate transformations of arbitrary Latin square. With this trick, we can generate the adjugates of arbitrary Latin square directly from the original one without generating the orthogonal array. The relations of isotopisms and adjugate 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 adjugate 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., Comment: Any comments and criticise are appreciated

The vertex $k$-partiteness $v_k(G)$ of graph $G$ is defined as the fewest number of vertices whose deletion from $G$ yields a $k$-partite graph. In this paper, we introduce two concepts: monotonic decreasing topological index and monotonic increasing topological index, and characterize the extremal graphs having the minimum Wiener index, the maximum Harry index, the maximum reciprocal degree distance, the minimum eccentricity distance sum, the minimum adjacent eccentric distance sum index, the maximum connective eccentricity index, the maximum Zagreb indices among graphs with a fixed number $n$ of vertices and fixed vertex $k$-partiteness, respectively., Comment: 15 pages

This paper is devoted to present two counterexamples to the theorem from \cite{MK} Maria R., Katherine T. M., Bernardo S. M., Extremal graphs with bounded vertex bipartiteness number, Linear Algebra Appl. 493 (2016) 28-36. Moreover, the corrected theorem and proof are presented.

The first multiplicative Zagreb index of a graph $G$ is the product of the square of every vertex degree, while the second multiplicative Zagreb index is the product of the degree of each edge over all edges. In our work, we explore the multiplicative Zagreb indices of bipartite graphs of order $n$ with diameter $d$, and sharp upper bounds are obtained for these indices of graphs in $\mathcal{B}(n,d)$, where $\mathcal{B}(n, d)$ is the set of all $n$-vertex bipartite graphs with the diameter $d$. In addition, we explore the relationship between the maximal multiplicative Zagreb indices of graphs \textcolor{blue}{within} $\mathcal{B}(n, d)$. As consequences, those bipartite graphs with the largest, second-largest and smallest multiplicative Zagreb indices are characterized, and our results extend and enrich some known conclusions., Comment: Accepted by Discrete Applied Mathematics

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., Comment: 19 pages, 25 figures, 15 tables

The hyper Zagreb index is a kind of extensions of Zagreb index, used for predicting physicochemical properties of organic compounds. Given a graph $G= (V(G), E(G))$, the first hyper-Zagreb index is the sum of the square of edge degree over edge set $E(G)$ and defined as $HM_1(G)=\sum_{e=uv\in E(G)}d(e)^2$, where $d(e)=d(u)+d(v)$ is the edge degree. In this work we define the second hyper-Zagreb index on the adjacent edges as $HM_2(G)=\sum_{e\sim f}d(e)d(f)$, where $e\sim f$ represents the adjacent edges of $G$. By inequalities, we explore some upper and lower bounds of these hyper-Zagreb indices, and provide the relation between Zagreb indices and hyper Zagreb indices., Comment: Accepted by Mathematical Reports

The Kirchhoff index of a graph is defined as half of the sum of all effective resistance distances between any two vertices. Assuming a complete multipartite graph G, by methods from linear algebra we explicitly formulate effective resistance distances between any two vertices of G, and its Kirchhoff index. In rest of paper we explore extremal value of Kirchhoff index for multipartite graphs