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Jia-Bao Liu

Publications and source records attributed to Jia-Bao Liu.

At least 37 records · Page 2Linked to original sources
arXiv2020

The (multiplicative degree-)Kirchhoff index of graphs derived from the Catersian product of $S_n$ and $K_2$

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 $S_n \times K_2$.

math.CO↗
arXiv2020

On the Maximum ABC Spectral Radius of Connected Graphs and Trees

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 $Δ=\max_{1\leq i \leq n} d_i$. 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 $ρ_{ABC}(G)$. Recently, this graph invariant has attracted some attentions. We prove that $ρ_{ABC}(G) \leq \sqrt{Δ+(2m-n+1)/Δ-2}$. As an application, the unique tree with $n \geq 4$ vertices having second largest ABC spectral radius is determined.

math.SP↗
arXiv2020

Metric dimension, minimal doubly resolving sets and the strong metric dimension for jellyfish graph and cocktail party graph

Let $Γ$ be a simple connected undirected graph with vertex set $V(Γ)$ and edge set $E(Γ)$. The metric dimension of a graph $Γ$ 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 $Γ$ and a vertex $v$ of $Γ$, 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 $Γ$ have different metric representations then the ordered set $W$ is called a resolving set of $Γ$. It is known that the problem of computing this invariant is NP-hard. In this paper, we consider the problem of determining the cardinality $ψ(Γ)$ of minimal doubly resolving sets of $Γ$, and the strong metric dimension for jellyfish graph $JFG(n, m)$ and cocktail party graph $CP(k+1)$.

math.GM↗
arXiv2020

The normalized Laplacians and random walks of the parallel subdivision graphs

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.

math.SP↗
arXiv2019

Resistance distance-based graph invariants and the number of spanning trees of linear crossed octagonal graphs

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.

math.SP↗
arXiv2018

On the largest $A_α$-spectral radius of cacti

Let $A(G)$ be the adjacent matrix and $D(G)$ the diagonal matrix of the degrees of a graph $G$, respectively. For $0 \leq α\leq 1$, the $A_α$ matrix $A_α(G) = αD(G) +(1-α)A(G)$ 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 $A_α$-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 Lovász and Pelikán. 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.

math.CO↗
arXiv2017

Graphs having extremal monotonic topological indices with bounded vertex $k$-partiteness

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.

math.CO↗
arXiv2017

Sharp upper bounds for multiplicative Zagreb indices of bipartite graphs with given diameter

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.

math.CO↗
arXiv2017

On extremal multiplicative Zagreb indices of trees with given number of vertices of maximum degree

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 products of degrees of pairs of adjacent vertices. In this paper, we explore the trees in terms of given number of vertices of maximum degree. The maximum and minimum values of $\prod_1(G) $ and $\prod_2(G) $ of trees with arbitrary number of maximum degree are provided. In addition, the corresponding extremal graphs are characterized.

math.CO↗
arXiv2017

A note on "Extremal graphs with bounded vertex bipartiteness number"

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.

math.CO↗
arXiv2016

Bounds of Zagreb indices and hyper Zagreb indices

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.

math.CO↗
arXiv2016

kirchhoff index and degree kirchhoff index of complete multipartite graph

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

math.CO↗
arXiv2016

On extremal multiplicative Zagreb indices of trees with given domination number

For a graph $G$, the first multiplicative Zagreb index $\prod_1$ is equal to the product of squares of the vertex degrees, and the second multiplicative Zagreb index $\prod_2$ is equal to the product of the products of degrees of pairs of adjacent vertices. The (mutiplicative) Zagreb indices have been the focus of considerable research in computational chemistry dating back to Gutman and Trinajstić in 1972. In this paper, we explore the mutiplicative Zagreb indices in terms of arbitrary domination number. The sharp upper and lower bounds of $\prod_1(G)$ and $\prod_2(G)$ are given. In addition, the corresponding extreme graphs are charaterized.

math.CO↗
arXiv2016

Resistance distance and Kirchhoff index in the corona-vertex and the corona $-$ edge of subdivision graph

The subdivision graph $S(G)$ of a graph $G$ is the graph obtained by inserting a new vertex into every edge of $G$. In $\cite{PL}$, two classes of new corona graphs, the corona-vertex of the subdivision graph $G_{1}\diamondsuit G_{2}$ and corona-edge of the subdivision graph $G_{1}\star G_{2}$ were defined. The adjacency spectrum and the signless Laplacian spectrum of the two new graphs were computed when $G_{1}$ is an arbitrary graph and $G_{2}$ is an $r$-regular graph. In this paper, we give the formulate of the resistance distance and the Kirchhoff index in $G_{1}\diamondsuit G_{2}$ and $G_{1}\star G_{2}$ when $G_{1}$ and $G_{2}$ are arbitrary graphs. These results generalize them in $\cite{PL}$.

math.CO↗
arXiv2016

Structure Properties of Koch Networks Based on Networks Dynamical Systems

We introduce an informative labeling algorithm for the vertices of a family of Koch networks. Each of the labels is consisted of two parts, the precise position and the time adding to Koch networks. The shortest path routing between any two vertices is determined only on the basis of their labels, and the routing is calculated only by few computations. The rigorous solutions of betweenness centrality for every node and edge are also derived by the help of their labels. Furthermore, the community structure in Koch networks is studied by the current and voltage characteristics of its resistor networks.

cs.SI↗
arXiv2016

Bicyclic graphs with extremal degree resistance distance

Let $r(u,v)$ be the resistance distance between two vertices $u, v$ of a simple graph $G$, which is the effective resistance between the vertices in the corresponding electrical network constructed from $G$ by replacing each edge of $G$ with a unit resistor. The degree resistance distance of a simple graph $G$ is defined as ${D_R}(G) = \sum\limits_{\{u,v\} \subseteq V(G)} {[d(u) + d(v)]r(u,v)},$ where $d(u)$ is the degree of the vertex $u$. In this paper, the bicyclic graphs with extremal degree resistance distance are strong-minded. We first determine the $n$-vertex bicyclic graphs having precisely two cycles with minimum and maximum degree resistance distance. We then completely characterize the bicyclic graphs with extremal degree resistance distance.

math.CO↗
arXiv2016

Further results on degree based topological indices of certain chemical networks

There are various topological indices such as degree based topological indices, distance based topological indices and counting related topological indices etc. These topological indices correlate certain physicochemical properties such as boiling point, stability of chemical compounds. In this paper, we compute the sum-connectivity index and multiplicative Zagreb indices for certain networks of chemical importance like silicate networks, hexagonal networks, oxide networks, and honeycomb networks. Moreover, a comparative study using computer-based graphs has been made to clarify their nature for these families of networks.

math.CO↗