Narrow Results

Let $G=(V_G,E_G)$ be a graph with an adjacency matrix $A_G$ and a diagonal degree matrix $D_G$. For any graph $G$ and a real number $\alpha \in [0,1]$, the $A_{\alpha }$-matrix of $G$, denoted by $A_{\alpha }(G)$, is defined as $A_{\alpha }(G)=\alpha D_G+(1-\alpha )A_G$. The generalized subdivision graph $S_G(n_1,m_1)$, derived from the subdivision graph of $G$ having the vertex set $V_G\cup E_G$, comprises a vertex set $V_G\times \{1,2,\dots ,n_1\}\cup E_G\times \{1,2,\dots ,m_1\}$. This construction includes $n_1$ replicas of $V_G$ and $m_1$ replicas of $E_G$, with edges established between vertices $(v,i)$ and $(e,j)$ where $e\in E_G$ is incident to $v\in V_G$ in $G$. In this paper, we derive the $A_{\alpha }$-characteristic polynomial of $S_G(n_1,m_1)$. We demonstrate that if $G$ is a regular graph, then the $A_{\alpha }$-spectrum of $S_G(n_1,m_1)$ is completely determined by the Laplacian spectrum of $G$. Specifically, when $n_1=m_1$, the $A_{\alpha }$-spectrum of $S_G(n_1,m_1)$ is completely determined by the Laplacian spectrum of the subdivision graph of $G$. In conclusion, as an application, we present the construction of infinite families of non-isomorphic graphs that are $A_{\alpha }$-cospectral.

In this paper, we investigate the random walks on metro systems in 28 cities from worldwide via the Laplacian spectrum to realize the trapping process on real systems. The average trapping time is a primary description to response the trapping process. Firstly, we calculate the mean trapping time to each target station and to each entire system, respectively. Moreover, we also compare the average trapping time with the strength (the weighted degree) and average shortest path length for each station, separately. It is noted that the average trapping time has a close inverse relation with the station’s strength but rough positive correlation with the average shortest path length. And we also catch the information that the mean trapping time to each metro system approximately positively correlates with the system’s size. Finally, the trapping process on weighted and unweighted metro systems is compared to each other for better understanding the influence of weights on trapping process on metro networks. Numerical results show that the weights have no significant impact on the trapping performance on metro networks.

Fermat–Wiener index based on topological Wiener index is the total sum of Fermat distance over all the triplets for vertices. In this paper, we construct a class of hierarchical graphs based on hierarchical product generalized from Cartesian product. We study some critical properties of the hierarchical networks by investigating its topological indices. Applying the finite pattern method, we analytically deduce the dominant term of average Fermat distance and obtain its asymptotic formula, which implies small-world property. We finally exhibit a close connection between Fermat–Wiener index and related graph invariants like average geodesic distance, Wiener index and eigenvalues of Laplacian matrix.

Laplacian spectrum gives a lot of useful information about complex structural properties and relevant dynamical aspects, which has attracted the attention of mathematicians. We introduced the weighted scale-free network inspired by the binary scale-free network. First, the weighted scale-free network with a weight factor is constructed by an iterative way. In the next step, we use the definition of eigenvalue and eigenvector to obtain the recursive relationship of its eigenvalues and multiplicities at two successive generations. Through analysis of eigenvalues of transition weight matrix we find that multiplicities of eigenvalues 0 of transition matrix are different for the binary scale-free network and the weighted scale-free network. Then, we obtain the eigenvalues for the normalized Laplacian matrix of the weighted scale-free network by using the obtained eigenvalues of transition weight matrix. Finally, we show some applications of the Laplacian spectrum in calculating eigentime identity and Kirchhoff index. The leading term of these indexes are completely different for the binary and the weighted scale-free network.

Weighted folded hypercube is an charming variance of the famous hypercube and is superior to the weighted hypercube in many criteria. We mainly study the scaling of network coherence for the weighted folded hypercube that is controlled by a weight factor. Network coherence quantifies the steady-state variance of these fluctuations, and it can be regarded as a measure of robustness of the consensus process to the additive noise. If networks with small steady-state variance have better network coherence, it can be regarded as more robust to noise than networks with low coherence. We firstly calculate the spectra of weighted folded hypercube and obtain the leading terms of network coherence that are quantified as the sum and square sum of reciprocals of all nonzero Laplacian eigenvalues. Finally, the results show that network coherence depends on iterations and weight factor. Meanwhile, with larger order, the scatings of the first- and second-order network coherence of weighted folded hypercube decrease with the increasing of weight factor.

In this paper, the synchronizability characterized by the Laplacian spectrum is applied to windmill networks where three types of parameters are introduced to control the number of deleted nodes. Using the network’s structures, exact solutions of the Laplacian eigenvalues are obtained and metrics of the synchronizability are correspondingly shown. Relationships between the synchronizability and the introduced parameters are presented. Then, the synchronizability of models with different settings of node failures is compared. The obtained results reveal distinct synchronizabilities originating from intrinsic structures of models and the setting forms of node failures. Finally, numerical examples are provided to verify the effectiveness of theoretical analysis.

Complex networks have attracted a great deal of attention from scientific communities, and have been proven as a useful tool to characterize the topologies and dynamics of real and human-made complex systems. Laplacian spectrum of the considered networks plays an essential role in their network properties, which have a wide range of applications in chemistry and others. Firstly, we define one vertex–vertex graph. Then, we deduce the recursive relationship of its eigenvalues at two successive generations of the normalized Laplacian matrix, and we obtain the Laplacian spectrum for vertex–vertex graph. Finally, we show the applications of the Laplacian spectrum, i.e. first-order network coherence, second-order network coherence, Kirchhoff index, spanning tree, and Laplacian-energy-like.

More and more attention has focused on consensus problem in the study of complex networks. Many researchers investigated consensus dynamics in a linear dynamical system with additive stochastic disturbances. In this paper, we construct iterated line graphs of multi-subdivision graph by applying multi-subdivided-line graph operation. It has been proven that the network coherence can be characterized by the Laplacian spectrum of network. We study the recursion formula of Laplacian eigenvalues of the graphs. After that, we obtain the scalings of the first- and second-order network coherence.

Recently, the consensus problem has attracted extensive attention of researchers from different scientific communities. In this paper, we study the leader–follower network coherence (i.e. the mean steady-state variance of the deviation from the static value of the leader nodes) in the weighted recursive tree networks with assigning the leaders orderly in the initial state. For the weighted recursive tree networks model, we study the characteristic polynomial of the Laplacian submatrix and obtain the relationship for the eigenvalues in two successive generations. The analytical formula of the leader–follower network coherence related by the sum of the reciprocal of all eigenvalues of this submatrix is obtained. The result shows that more number of leaders and the greater weight factor lead to better consensus.

The power graph ${𝒢}(S)$ of a semigroup $S$ is a simple undirected graph whose vertex set is $S$ itself, and any two distinct vertices are adjacent if one of them is a power of the other. In this paper, we describe the power graph ${𝒢}({\mathbb{Z}}_p^r\times {\mathbb{Z}}_{p^2}^s)$ in terms of joins and disjoint unions of complete graphs, and use this to calculate the Laplacian polynomial of ${𝒢}({\mathbb{Z}}_p^r\times {\mathbb{Z}}_{p^2}^s)$. We finally calculate the Laplacian polynomial of the power graph ${𝒢}({\mathbb{Z}}_{p^m}^n)$.

In this paper, we introduce a graph product, namely, $M$-bicone product and determine the generalized characteristic polynomial of the graphs obtained from this product. Consequently, we obtain the characteristic polynomial of the adjacency matrix, the Laplacian matrix and the signless Laplacian matrix of this graph. Also, from these results, we obtain the $L$-spectra of some families of bicone product of graphs. As an application, we obtain infinite pairs of $L$-cospectral graphs.

In the past decades, graphs that are determined by their spectrum have received much more and more attention, since they have been applied to several fields, such as randomized algorithms, combinatorial optimization problems and machine learning. An important part of spectral graph theory is devoted to determining whether given graphs or classes of graphs are determined by their spectra or not. So, finding and introducing any class of graphs which are determined by their spectra can be an interesting and important problem. The main aim of this study is to characterize two classes of multicone graphs which are determined by their adjacency, Laplacian and signless Laplacian spectra. A multicone graph is defined to be the join of a clique and a regular graph. Let $K_w$ denote a complete graph on $w$ vertices. In the paper, we show that multicone graphs $K_w▽LHS$ and $K_w▽LGQ(3,9)$ are determined by both their adjacency spectra and their Laplacian spectra, where $LHS$ and $LGQ(3,9)$ denote the Local Higman–Sims graph and the Local $GQ(3,9)$ graph, respectively. In addition, we prove that these multicone graphs are determined by their signless Laplacian spectra.

In this paper, we define some variants of corona of graphs namely, subdivision (respectively, $R$-graph, $Q$-graph, total) neighborhood corona, $R$-graph (respectively, $Q$-graph, total) semi-edge neighborhood corona, $R$-graph (respectively, total) semi-vertex neighborhood corona of graphs constrained by vertex subsets. These corona operations generalize some existing corona operations such as subdivision ($R$-graph, $Q$-graph, total) double neighborhood corona, subdivision vertex (respectively, edge) neighborhood corona, $R$-graph vertex (respectively, edge) neighborhood corona of graphs. First, we consider a matrix in specific form and determine its spectrum. Then by using this, we derive the characteristic polynomials of the adjacency and the Laplacian matrices of the new graphs when the base graph is regular. Also, we deduce the characteristic polynomials of the adjacency and Laplacian matrices of the above mentioned particular cases from our results.

In this paper, we introduce a new graph product, namely, bowtie product. We determine the generalized characteristic polynomial of the bowtie product of graphs and consequently, we obtain the characteristic polynomial of the adjacency martix, the Laplacian matrix, the signless Laplacian matrix and the normalized Laplacian matrix of this graph. Also, from these results, we obtain the $L$-spectra of some families of bowtie product of graphs. As an application, we obtain infinite pairs of $L$-cospectral graphs and construct infinite number of $L$-integral graphs.

Let G be a simple undirected graph. Three new corona products of graphs based on splitting graph of G are defined. The adjacency spectra of the three new graphs based on splitting graph of G are determined. The number of spanning trees and the Kirchoff index of the new graphs are determined using their nonzero Laplacian eigenvalues.

The enhanced power graph of a group $G$ is a simple undirected graph with vertex set $G$ and two vertices are adjacent if they belong to the same cyclic subgroup. In this paper, we obtain the Laplacian spectrum of the enhanced power graph of certain non-abelian groups, viz. semidihedral, dihedral and generalized quaternion. Also, we obtained the metric dimension and the resolving polynomial of the enhanced power graphs of these groups. At the final part of this paper, we study the distant properties and the detour distant properties, namely: closure, interior, distance degree sequence, eccentric subgraph of the enhanced power graph of semidihedral group, dihedral group and generalized quaternion group, respectively.