Extended Adjacency Matrix Research Articles

On spectral radius of generalized arithmetic–geometric matrix of connected graphs and its applications

The generalized arithmetic–geometric matrix of a simple connected graph G=(V,E) is defined to be the |V|×|V| matrix whose ij-entry is diα+djα2diαdjα if vivj∈E, and 0 otherwise, where α is an arbitrary real number and di is the degree of vi∈V. This matrix is a general form of the arithmetic–geometric matrix (α=1) and the extended adjacency matrix (α=2), both of which have been well studied in spectral graph theory and chemical graph theory. In this paper, we focus on the spectral radius ρagα of the generalized arithmetic–geometric matrix of connected graphs. Some chemical applications of ρagα are explored, and some extremal results on ρagα are obtained; in particular, the connected graphs (including trees) having the maximum and minimum ρagα are determined.

Retraction Note: The Sachs theorem and its application on extended adjacency matrix of graphs

Retraction Note: The Sachs theorem and its application on extended adjacency matrix of graphs

Development and uniqueness test of highly selective atomic topological indices based on the number of attached hydrogen atoms

On the basis of the atomic graph-theoretical index – aEAID (atomic Extended Adjacency matrix IDentification) and molecular adjacent topological index – ATID (Adjacent Topological IDentification) suggested by one of the authors (Zhang Q), a highly selective atomic topological index - aATID (atomic Adjacent Topological IDentification) index was suggested to identify the equivalent atoms in this study. The aATID index of an atom was derived from the number of the attached hydrogen atoms of the atom but omitting bond types. In this case, the suggested index can be used to identify equivalent atoms in chemistry but perhaps not equivalent in the molecular graph. To test the uniqueness of aATID indices, the virtual atomic data sets were derived from alkanes containing 15–20 carbon atoms and the isomers of Octogen, as well as a real data set was derived from the NCI database. Only four pairs of atoms from alkanes containing 20 carbons can't be discriminated by aATID, that is, four pairs of degenerates were found for this data set. To solve this problem, the aATID index was modified by introducing distance factors between atoms, and the 2-aATID index was suggested. Its uniqueness was examined by 5,939,902 atoms derived from alkanes containing 20 carbons and further 16,166,984 atoms from alkanes of 21 carbons, and no degenerates were found. In addition, another large real data set of 16,650,688 atoms derived from the PubChem database was also used to test the uniqueness of both aATID and 2-aATID. As a result, each atom was successfully discriminated by any of the two indices. Finally, the suggested aATID index was applied to the identification of duplicate atoms as data pretreatment for QSPR (Quantitative Structure–Property Relationships) studies.

Degree based energy and spectral radius of a graph with self-loops

Let GX be a graph obtained from a simple graph G by attaching a self-loop at each vertex of X ⊆ V ( G ) . The general extended adjacency matrix for the graph GX is defined and the bounds for the degree based energy of the graph GX are obtained. The study extends the notion of degree based energy of simple graphs to graphs with self-loops. For the graph GX of order n and size m with σ self-loops, the adjacency energy, E ( G X ) ≥ 4 mn + 2 σ ( n − σ ) n . The spectral radius ρ ( G X ) of its adjacency matrix is always less than or equal to 1 2 [ − 1 + 1 + 4 ( 2 m + σ + Δ − 1 ) ] , where Δ is the maximum degree in the graph GX and the equality conditions are given for 0 < σ < n . Few more bounds for ρ ( G X ) are also obtained. The study shows that, the spectral radius f max ( G X ) of its extended adjacency matrix satisfies: min i ∼ j F ij ( 2 m + σ ) n ≤ f max ( G X ) ≤ max i ∼ j F ij 2 ( 2 m + σ ) . We conclude the article by computing the extended adjacency spectrum of complete graph and complete bipartite graphs with self-loops.

The General Extended Adjacency Eigenvalues of Chain Graphs

In this article, we discuss the spectral properties of the general extended adjacency matrix for chain graphs. In particular, we discuss the eigenvalues of the general extended adjacency matrix of the chain graphs and obtain its general extended adjacency inertia. We obtain bounds for the largest and the smallest general extended adjacency eigenvalues and characterize the extremal graphs. We also obtain a lower bound for the spread of the general extended adjacency matrix. We characterize chain graphs with all the general extended adjacency eigenvalues being simple and chain graphs that are non-singular under the general extended adjacency matrix. Further, we determine the explicit formula for the determinant and the trace of the square of the general extended adjacency matrix of chain graphs. Finally, we discuss the energy of the general extended adjacency matrix and obtain some bounds for it. We characterize the extremal chain graphs attaining these bounds.

Friendship Inference in Mobile Social Networks: Exploiting Multi-Source Information With Two-Stage Deep Learning Framework

With the tremendous growth of mobile social networks (MSNs), people are highly relying on it to connect with friends and further expand their social circles. However, the conventional friendship inference techniques have issues handling such a large yet sparse multi-source data. The related friend recommendation systems are therefore suffering from reduced accuracy and limited scalability. To address this issue, we propose a Two-stage Deep learning framework for Friendship Inference, namely TDFI. This approach enables MSNs to exploit multi-source information simultaneously, rather than hierarchically. Therefore, there is no need to manually set which information is more important and the order in which the various information is applied. In details, we apply an Extended Adjacency Matrix (EAM) to represent the multi-source information. We then adopt an improved Deep Auto-Encoder Network (iDAEN) to extract the fused feature vector for each user. Our framework also provides an improved Deep Siamese Network (iDSN) to measure user similarity. To provide a substantial description and evaluation of the proposed methodology, we evaluate the effectiveness and robustness on three large-scale real-world datasets. Trace-driven evaluation results demonstrate that TDFI can effectively handle the sparse multi-source data while providing better accuracy for friendship inference. Through the comparison with numerous state-of-the-art methods, we find that TDFI can achieve superior performance via real-world multi-source information. Meanwhile, it demonstrates that the proposed pipeline can not only integrate structural information and attribute information, but also be compatible with different attribute information, which further enhances the overall applicability of friend-recommendation systems under information-rich MSNs.

The Extended Adjacency Indices for Several Types of Graph Operations

Let G be a simple graph without multiple edges and any loops. At first, the extended adjacency matrix of a graph was first proposed by Yang et al in 1994, which is explored from the perspective of chemical molecular graph. Later, the spectral radius of graph and graph energy under the extended adjacency matrix was proposed. At the same time, for a simple graph G, the extended adjacency index <I>EA(G)</I> is also defined by some researchers. All of them play important roles in mathematics and chemistry. In this work, we show the extended adjacency indices for several types of graph operations such as tensor product, disjunction and strong product. In addition, we also give some examples of different combinations of special graphs, such as complete graphs and cycle graphs, and the classical graph, Cayley graph. By combining the special structure of the graph, it will pave the way for the calculation of some chemical or biological classical molecular structure. We can find that it plays a meaningful role in calculating the structure of complex chemical molecules through the graph operation of EA index on the any simple combined graphs, and it can also play a role in biology, physics, medicine and so on. Finally, we put forward some other related problems that can be further studied in the future.

A note on eigenvalue, spectral radius and energy of extended adjacency matrix

The extended adjacency matrix of a graph has been introduced to expand some beneficial molecular topological indices. In the present work, the spectra of certain classes of vertex or edge-transitive graphs are investigated. Also, we study some spectral properties of extended adjacency matrix. Besides, we study the spectral A e x -spread of a graph. In this way, we explore some new bounds for both the smallest and the largest eigenvalues. Finally, we persue the behaviour of A e x -energy of a graph.

The Sachs theorem and its application on extended adjacency matrix of graphs

The Sachs theorem and its application on extended adjacency matrix of graphs

The weighted sitting closer to friends than enemies problem in the line

The weighted Sitting Closer to Friends than Enemies (SCFE) problem is to find an injection of the vertex set of a given weighted graph into a given metric space so that, for every pair of incident edges with different weight, the end vertices of the heavier edge are closer than the end vertices of the lighter edge. The Seriation problem is to find a simultaneous reordering of the rows and columns of a symmetric matrix such that the entries are monotone nondecreasing in rows and columns when moving towards the diagonal. If such a reordering exists, it is called a Robinson ordering. In this work, we establish a connection between the SCFE problem and the Seriation problem. We show that if the extended adjacency matrix of a given weighted graph G has no Robinson ordering then G has no injection in $$\mathbb {R}$$ that solves the SCFE problem. On the other hand, if the extended adjacency matrix of G has a Robinson ordering, we construct a polyhedron that is not empty if and only if there is an injection of the vertex set of G in $$\mathbb {R}$$ that solves the SCFE problem. As a consequence of these results, we conclude that deciding the existence of (and constructing) such an injection in $$\mathbb {R}$$ for a given complete weighted graph can be done in polynomial time. On the other hand, we show that deciding if an incomplete weighted graph has such an injection in $$\mathbb {R}$$ is NP-Complete.

A relation between a vertex-degree-based topological index and its energy

Let G be a simple graph with vertex set V and edge set E, and let di be the degree of the vertex vi∈V. If the vertices vi and vj are adjacent, we denote the respective edge by vivj∈E. A vertex-degree-based topological index φ is defined as φ(G)=∑vivj∈Eφdi,dj, where φi,j is a function with the property φi,j=φj,i. The general extended adjacency matrix Aφ is defined as [Aφ]ij=φdi,dj if vivj∈E, and 0 otherwise. The energy associated to φ of G is the sum of the absolute values of the eigenvalues of Aφ.In this paper we show that ρ(G)Eφ(G)≥2φ(G) for all connected graphs G, where ρ(G) is the spectral radius of G and φa,b≠0 for all a,b∈N. We also characterize the graphs where equality holds. As a consequence, for any tree T with n vertices, Eφ(T)≥2n−1φ1,(n−1), with equality holding if and only if T≅Sn.

On the eigenvalue and energy of extended adjacency matrix

The extended adjacency matrix of graph G,Aex is a symmetric real matrix that if i≠j and uiuj∈E(G), then the ijth entry is dui2+duj2/2duiduj, and zero otherwise, where du indicates the degree of vertex u. In the present paper, several investigations of the extended adjacency matrix are undertaken and then some spectral properties of Aex are given. Moreover, we present some lower and upper bounds on extended adjacency spectral radii of graphs. Besides, we also study the behavior of the extended adjacency energy of a graph G.

SOME NOTES ON THE LAPLACIAN ENERGY OF EXTENDED ADJACENCY MATRIX

The extended adjacency matrix Ae(G) of a graph G is determined by graph degrees. Some inequalities are found for the extended adjacency matrix including its vertices, its edges and its degrees in this paper. Also, some bounds are established for this matrix involving its energy and its Laplacian energy.

A data-driven robust optimization method for the assembly job-shop scheduling problem under uncertainty

ABSTRACT This paper studies the production scheduling problem in an assembly manufacturing system with uncertain processing time and random machine breakdown. The objectives of minimizing makespan and the performance deviation of the actual schedule from the baseline schedule are simultaneously considered. Specifically, a boosting radial basis function network constructed using the data generated by Monte Carlo method, is used as the surrogate model to approximate the performance deviation. After that, a modified master-apprentice evolutionary algorithm (MAE) is developed for robust scheduling. In the design of MAE, we employ an extended adjacency matrix of subassemblies to cope with the sequential constraints of operations in AJSSP. Based on this, effective neighbourhood structures and distance metric of solutions are designed for tabu search and path relinking operators to generate feasible schedules. To evaluate the effectiveness of the proposed method, a series of computational experiments are conducted. The results indicate that, compared with several commonly used algorithms, the suggested method shows good performance in dealing with AJSSP under uncertainty.

Schematic Diagrams Design of Displacement Suppression Mechanism with One Degree-of-Freedom in a Rope-Guided Hoisting System

Since it is difficult for lateral stiffness of rope-guided rails to meet industry criteria in deep construction shaft, schematic diagrams of displacement suppression mechanisms (DSMs) are designed with a systematic approach demonstrated to reduce the lateral displacement of rope-guided rails in this paper. DSMs are simplified as planar four-bar and six-bar topological graphs based on topological theory. Each corresponding mechanical chain of these four-bar and six-bar mechanisms is divided into a rack, mechanical parts, prismatic, and revolute joints. An extended adjacency matrix is defined to represent the rack position, specific types of kinematic joints, and adjacency relationships between kinematic parts. Then, a symmetric vertex identification method is proposed with regard to planar 1-DOF (one degree of freedom) four-bar and six-bar topological graphs to get the sequences of prismatic joints for kinematic chains of DSMs. Finally, the alternative schematic diagrams of DSMs are obtained. The results show four-bar mechanisms with simple structure; few kinematical parts but less resident force are suitable for a mine shaft with small space and small swing. Six-bar mechanisms with two prismatic joints and three mechanical rack degree are applicable for wide shaft space in deep shaft, due to their stable structure and double resistant force. This development is helpful for DSM dimension synthesis design in future.

Bounds for the spectral radius and energy of extended adjacency matrix of graphs

ABSTRACT An extend adjacency matrix of a graph () was introduced decades ago as a precursor for developing a few quite useful molecular topological descriptors. The spectral radius () of the extended adjacency matrix and the extended energy of a graph () have been successfully utilized in QSPR/QSAR investigations. Here, the and have been further mathematically analyzed. Several sharp upper bounds for the are obtained. In addition, the Nordhaus–Gaddum-type results for the and are presented.

On the extended Estrada index of some graphs

Let G be a graph on n vertices and η1,η2,…,ηn the eigenvalues of its extended adjacency matrix. The extended Estrada index EEex is defined as the sum of the terms eηi,i=1,2,…,n. In this paper we show the extended Estrada index of the complete bipartite graphs and the friendship graphs. Also the relationship of the extended Estrada index between the m-splitting, m-shadow graphs and the original graphs are obtained.

Power Grid Partitioning Based on Functional Community Structure

Network partitioning is a popular research topic. Not all available partitioning methods are equally suitable for power grids. Community detection is a critical issue in complex network theory, and power grid is a typical type of complex network. This paper proposes a functional community structure based on an extended weighted network model. An extended adjacency matrix is used to represent an extended weighted complex network model based on coupling strength rather than the conventional adjacency matrix. Meanwhile, we upgraded the Newman fast algorithm of community detection for establishing a novel power grid partitioning algorithm. The electrical coupling strength (ECS) is defined to better reflect electrical characteristics between any two nodes in power grid. Modularity is also redefined as electrical modularity based on ECS. The Newman fast algorithm is upgraded with electrical modularity maximization as the objective to detect functional communities in power grids. A case study on IEEE test systems with 30, 39, 118, 300 buses and one Italian power network demonstrates the rationality of the extended weighted network model and partitioning algorithm.

Degree-based energies of graphs

Let G=(V,E) be a simple graph of order n and size m, with vertex set V(G)={v1,v2,…,vn}, without isolated vertices and sequence of vertex degrees Δ=d1≥d2≥⋯≥dn=δ>0, di=dG(vi). If the vertices vi and vj are adjacent, we denote it as vivj∈E(G) or i∼j. With TI we denote a topological index that can be represented as TI=TI(G)=∑i∼jF(di,dj), where F is an appropriately chosen function with the property F(x,y)=F(y,x). A general extended adjacency matrix A=(aij) of G is defined as aij=F(di,dj) if the vertices vi and vj are adjacent, and aij=0 otherwise. Denote by fi, i=1,2,…,n the eigenvalues of A. The “energy” of the general extended adjacency matrix is defined as ETI=ETI(G)=∑i=1n|fi|. Lower and upper bounds on ETI are obtained. By means of the present approach a plethora of earlier established results can be obtained as special cases.

Upper bounds for the extended energy of graphs and some extended equienergetic graphs

In this paper, we give two upper bounds for the extended energy of a graph one in terms of ordinary energy, maximum degree and minimum degree of a graph, and another bound in terms of forgotten index, inverse degree sum, order of a graph and minimum degree of a graph which improves an upper bound of Das et al. from [On spectral radius and energy of extended adjacency matrix of graphs, Appl. Math. Comput. 296 (2017), 116-123]. We present a pair of extended equienergetic graphs on \(n\) vertices for \(n\equiv 0(\text{mod } 8)\) starting with a pair of extended equienergetic non regular graphs on \(8\) vertices and also we construct a pair of extended equienergetic graphs on \(n\) vertices for all \(n\geq 9\) starting with a pair of equienergetic regular graphs on \(9\) vertices.