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

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.

Connectivity of vertex and edge transitive graphs

The adjacency matrix operations,which connect with configuration transformation correspondingly,can be used for analysis of configuration transformation of metamorphic mechanisms and the corresponding algorithm can easily be simulated by computer.But the adjacency matrix based on monochrome topological graph is not suitable for the topological representation of mechanisms with multiple joints.The method of adjacency matrix operations has its own limitations for analysis of configuration transformation of metamorphic mechanisms because it can only be used in the topological representation of mechanisms with single joints.In order to overcome the drawback of the adjacency matrix,a kind of new matrix named as extended adjacency matrix is proposed to express topological structures of all mechanisms.The extended adjacency matrix is not only suitable for the topological representation of mechanisms with single joints,but also can be used in that of mechanisms with multiple joints.On this basis,a method of matrix operations based on the extended adjacency matrix is proposed to analyze the configuration transformation of metamorphic mechanisms.The method is not only suitable for configuration analysis of metamorphic mechanisms with single joints as well as metamorphic mechanisms with multiple joints.The method is evaluated by calculating two examples representing metamorphic mechanisms with single joint and multiple joints respectively.It can be concluded that the method is effective and correct for analysis of configuration transformation of all metamorphic mechanisms.The proposed method is simple and easy to be achieved by computer programming.It provides a basis for structural synthesis of all metamorphic mechanisms.

The spectral radius of a network graph is the largest Eigen value of the adjacency matrix of the graph. We hypothesize the spectral radius to be a measure of the variation in the degrees of the nodes. In this pursuit, we define a metric called the spectral radius ratio for node degree as the ratio of the spectral radius to the average node degree. We validate our hypothesis by determining this metric on some of the commonly studied classical large real-world complex network graphs (undirected) for network analysis. Based on the results collected, we observe the spectral radius ratio for node degree to be positively correlated (correlation coefficient: 0.75) to the coefficient of variation in node degree (the ratio of the average node degree to the standard deviation in node degree), thus confirming our hypothesis.

Ordering graphs with index in the interval [formula omitted

For 0 ? ? ? 1, Nikiforov proposed to study the spectral properties of the family of matrices A?(G) = ?D(G)+(1 ? ?)A(G) of a graph G, where D(G) is the degree diagonal matrix and A(G) is the adjacency matrix of G. The ?-spectral radius of G is the largest eigenvalue of A?(G). For a graph with two pendant paths at a vertex or at two adjacent vertices, we prove results concerning the behavior of the ?-spectral radius under relocation of a pendant edge in a pendant path. We give upper bounds for the ?-spectral radius for unicyclic graphs G with maximum degree ? ? 2, connected irregular graphs with given maximum degree and some other graph parameters, and graphs with given domination number, respectively. We determine the unique tree with the second largest ?-spectral radius among trees, and the unique tree with the largest ?-spectral radius among trees with given diameter. We also determine the unique graphs so that the difference between the maximum degree and the ?-spectral radius is maximum among trees, unicyclic graphs and non-bipartite graphs, respectively.

Network similarity measures have proven essential in the field of network analysis. Also, topological indices have been used to quantify the topology of networks and have been well studied. In this paper, we employ a new topological index which we call the Ediz eccentric connectivity index. We use this quantity to define network similarity measures as well. First, we determine the extremal value of the Ediz eccentric connectivity index on some network classes. Second, we compare the network similarity measure based on the Ediz eccentric connectivity index with other well-known topological indices such as Wiener index, graph energy, Randić index, the largest eigenvalue, the largest Laplacian eigenvalue, and connectivity eccentric index. Numerical results underpin the usefulness of the chosen measures. They show that our new measure outperforms all others, except the one based on Wiener index. This means that the measure based on Wiener index is still the best, but the new one has certain advantage to some extent.

In geographic, the eigenvalues and eigenvectors of transportation network provides many informations about its connectedness. It is proven that the more highly connected in a transportation network G has largest eigenvalue and hence more multiple occurrences of the eigenvalue -1. For a graph G with adjacency matrix A, the multiplicity of the eigenvalue -1 equals the dimension of the null space of the matrix A + I. In this paper, we constructed a high closed zero sum weighting of G and by which its proved that, the dimension of the null space of the eigenvalue -1 is the same as the number of independent variables used in a non-trivial high closed zero sum weighting of the graph. Multiplicity of -1 as an eigenvalue of known graphs and of corona product of certain classes of graphs are determined and two classes of -1- nut graphs are constructed.

Due to the explosive growth of social network services, friendship inference has been widely adopted by Online Social Service Providers (OSSPs) for friend recommendation. The conventional techniques, however, have limitations in accuracy or scalability to handle such a large yet sparse multi-source data. For example, the OSSPs will be required to manually give the order in which the various information is applied. This unavoidably reduces the applicability of existing friend recommendation systems. To address this issue, we propose a Two-stage Deep learning framework for Friendship Inference (TDFI). This approach can utilize multi-source information simultaneously with low complexity. In particular, we apply an Extended Adjacency Matrix (EAM) to represent the multi-source information. We then adopt an improved Deep AutoEncoder Network (iDAEN) to extract the fused feature vector for each user. The TDFI framework also provides an improved Deep Siamese Network (iDSN) to measure user similarity from iDAEN. Finally, we evaluate the effectiveness and robustness of TDFI on three large-scale real-world datasets. It shows that TDFI can effectively handle the sparse multi-source data while providing better accuracy for friend recommendation.

Connected hypergraphs with small spectral radius

Let μ(G) denote the spectral radius of a graph G. We partly confirm a conjecture due to Nikiforov, which is a spectral radius analogue of the well-known Erdős-Sós Conjecture that any tree of order t is contained in a graph of average degree greater than t−2. Let Sn,k be the graph obtained by joining every vertex of a complete graph on k vertices to every vertex of an independent set of order n−k, and let Sn,k+ be the graph obtained from Sn,k by adding a single edge joining two vertices of the independent set of Sn,k. In 2010, Nikiforov conjectured that for a given integer k, every graph G of sufficiently large order n with μ(G)≥μ(Sn,k+) contains all trees of order 2k+3, unless G=Sn,k+. We confirm this conjecture for trees with diameter at most four, with one exception. In fact, we prove the following stronger result for k≥8. If a graph G with sufficiently large order n satisfies μ(G)≥μ(Sn,k) and G≠Sn,k, then G contains all trees of order 2k+3 with diameter at most four, except for the tree obtained from a star on k+2 vertices by subdividing each of its k+1 edges once.

Many practical problems (frequency assignement, register allocation, timetables) may be formulated as graph (vertex-)coloring problems, but finding solutions for them is difficult as long as they are treated in the most general case (where the graph is arbitrary), since vertex coloring has been proved to be NP-complete. The problem becomes much easier to solve if the graph resulting from the modelisation of the practical application belongs to some particular class of graphs, for which solutions to the problem are known. Meyniel graphs form such a class (a fast coloring algorithm can be found in [9]), for which an efficient recognizing algorithm is therefore needed. A graph G=(V,E) is said to be a Meyniel graph if every odd cycle of G on at least five vertices contains at least two chords. Meyniel graphs generalize both i-triangulated and parity graphs, two well known classes of perfect graphs that will be present in our paper in Section 7. In [2], Burlet and Fonlupt propose a characterization of Meyniel graphs which relies on the following property: the class of Meyniel graphs may be obtained from some basic Meyniel graphs using a binary operation called amalgam. Besides the theoretical interest of this result, a practical interest arises because of the polynomial recognition algorithm which can be obtained. Unfortunately, it is quite expensive to verify if a given graph is the amalgam of two graphs (therefore the complexity of the whole algorithm is in O(n7)), and this supports the idea that a new point of view is needed to find a more efficient algorithm. Our approach of Meyniel graphs will be directed through the search of a general structure. Intuitively, a Meyniel graph either will be simple (i.e. with no hole or domino), or will have a skeleton around which the rest of the graph will be regularly organized. As suggested, the first type of Meyniel graphs is simple to identify. For the second type, a deeper analysis is necessary; it yields a characterization theorem, which is used to deduce the O(m2+mn) recognition algorithm.

Convex and quasiconvex functions on trees and their applications

On the spectral radii and the signless Laplacian spectral radii of c-cyclic graphs with fixed maximum degree

On spectral extrema of graphs with given order and dissociation number