Graph Spectrum

Let G be a graph with n vertices and m edges. Let λ 1 , λ 2 , . . . , λ n be the eigenvalues of the adjacency matrix of G, and let µ 1 , µ 2 , . . . , µ n be the eigenvalues of the Laplacian matrix of G. An earlier much studied quantity E(G) = n i=1 |λ i | is the energy of the graph G. We now define and investigate the Laplacian energy as LE(G) = n i=1 |µ i − 2m/n|. There is a great deal of analogy between the properties of E(G) and LE(G), but also some significant differences.

Due to the proliferation of drafting packages, there are a lot of vector drawings available for people to integrate into documents. Moreover, creative designers and drafts-people often reuse data from previous projects and libraries of ready-to-use components. Usually, retrieving these drawings is a slow, complex and error-prone endeavor. While text-driven attempts at classifying image data have been recently supplemented with query-by-image content, these work only for bitmap-type data and cannot handle vectorial information. Furthermore, they are not very good at retrieving partial information, which makes it difficult to search complex documents. Here we present a new approach to allow efficient retrieval of complex vector drawings by content using simple sketches as queries. Our work uses a new hierarchical topology description mechanism, which allows comparing complex drawings to simple queries. In particular, we produce a multilevel description of drawings, from the topology graph, using different levels of detail and subparts of the information. We also show that the graph spectrum is stable and reliable for use as a topology description. Finally, experimental tests show that our approach is good at retrieving complex drawings from two different application domains, CAD and clip-art figures.

Numerical comparison of spaces with one another is often achieved with set scalar measures such as global and local integration, connectivity, etc., which capture a particular quality of the space but therefore lose much of the detail of its overall structure. More detailed methods such as graph edit distance are difficult to calculate, particularly for large plans. This paper proposes the use of the graph spectrum, or the ordered eigenvalues of a graph adjacency matrix, as a means to characterise the space as a whole. The result is a vector of high dimensionality that can be easily measured against others for detailed comparison. Several graph types are investigated, including boundary and axial representations, as are several methods for deriving the spectral vector. The effectiveness of these is evaluated using a genetic algorithm optimisation to generate plans to match a given spectrum, and evolution is seen to produce plans similar to the initial targets, even in very large sea...

Due to the proliferation of drafting packages, there are a lot of vector drawings available for people to integrate into documents. Moreover, creative designers and drafts-people often reuse data from previous projects and libraries of ready-to-use components. Usually, retrieving these drawings is a slow, complex and error-prone endeavor. While text-driven attempts at classifying image data have been recently supplemented with query-by-image content, these work only for bitmap-type data and cannot handle vectorial information. Furthermore, they are not very good at retrieving partial information, which makes it difficult to search complex documents.

The energy of a graph G is the sum of the absolute values of the eigenvalues of the adjacency matrix of G. The Laplacian (respectively, the signless Laplacian) energy of G is the sum of the absolute values of the differences between the eigenvalues of the Laplacian (respectively, signless Laplacian) matrix and the arithmetic mean of the vertex degrees of the graph. In this paper, among some results which relate these energies, we point out some bounds to them using the energy of the line graph of G. Most of these bounds are valid for both energies, Laplacian and signless Laplacian. However, we present two new upper bounds on the signless Laplacian which are not upper bounds for the Laplacian energy.

For an arbitrary simple graph G and a positive integer r, the super line multigraph of index r of G, denoted Mr(G), has for vertices all the r-subsets of edges. Two vertices S and T are joined by as many edges as pairs of distinct edges s in S and t in T share a common vertex in G. We present spectral properties of Mr(G) and particularly, if G is a regular graph, we calculate all the eigenvalues of Mr(G) and their multiplicities in terms of those of G.

This paper proposes a novel non-parametric technique for clustering networks based on their structure. Many topological measures have been introduced in the literature to characterize topological properties of networks. These measures provide meaningful information about the structural properties of a network, but many networks share similar values of a given measure . Furthermore, strong correlation between these measures occur on real-world graphs [2], so that using them to distinguish arbitrary graphs is difficult in practice .

In this paper, we introduce a new measure of symme- try which we call the symmetry ratio of a network, de- flned to be the ratio of the number of distinct eigenval- ues of the network to D+1, where D is the diameter. The symmetry ratio has utility in partially predicting the robustness of a network in the face of

The energy of a graph G is equal to the sum of the absolute values of the eigenvalues of G, which in turn is equal to the sum of the singular values of the adjacency matrix of G. Let X, Y, and Z be matrices, such that X + Y = Z. The Ky Fan theorem establishes an inequality between the sum of the singular values of Z and the sum of the sum of the singular values of X and Y. This theorem is applied in the theory of graph energy, resulting in several new inequalities, as well as new proofs of some earlier known inequalities.

The local spectrum of a graph G = (V, E), constituted by the standard eigenvalues of G and their local multiplicities, plays a similar role as the global spectrum when the graph is "seen" from a given vertex. Thus, for each vertex i ∈ V , the i-local multiplicities of all the eigenvalues add up to 1; whereas the multiplicity of each eigenvalue λ of G is the sum, extended to all vertices, of its local multiplicities.

Currently, there are large collections of clip-art vector drawings from which users can select the desired figures to insert in their documents. However, to locate a particular drawing among thousands is not easy. Although there are some solutions for drawing retrieval, almost all of them are designed to retrieve simple and not complex drawings as for instance clip-arts. In our prior work we proposed an approach to index and retrieve complex vector drawings by content, using topological and geometric information automatically extracted from figures. In this paper, we present a new algorithm to improve the geometric matching of two drawings, which takes into account the drawing as a whole and each of the shapes in it. We developed a web search engine for clip-art drawings, where we included this new technique. Experimental evaluation reveals that this new geometric matching algorithm conducts to better retrieval results than the prior matching solution.

Let G be a graph on n vertices v 1 ,. .. , v n and let d(v i) be the degree of the vertex v i. If d(G) = (d(v 1),. .. , d(v n)) is an eigenvector of the (0, 1)-adjacency matrix A of G, i.e. A(G)d(G) = λd(G), then G is said to be λ-harmonic. In this paper all connected integral 3-harmonic graphs are determined. There are exactly 26 such graphs.

We present a low-complexity, accurate and robust localization algorithm suitable for large scale Wireless Sensor Networks (WSNs). The algorithm is a clusterized version of the Weighted Least-Squares (WLS) localization technique which we recently introduced in [1]. The WLS algorithm is a lowcomplexity localization technique that owes its high-accuracy to the ability to complete and approximate the Euclidean Distance Matrix (EDM) samples constructed from incomplete and errordisturbed ranging information collected from the sensors. The performance of this algorithm is, however, known to decrease sharply [1] when the completeness is not sufficient to ensure the uniqueness of the network (Graph) realization [2]. The clusterization procedure is based on recent Graph-theoretical results showing that the elements of the second smallest eigenvector of the Laplacian matrix of a Graph are strongly correlated with the proximity of its vertices. This Graphspectrum analytical tool is utilized here to separate the network into sub-groups that satisfy the completeness constraints of the WLS technique. The resulting clusterization procedure, which relies solely on connectivity information, allows the WLS to be applied into smaller parts of the network, each exhibiting a prescribed completeness level, leading simultaneously to a significant improvement in accuracy and to a reduction in the computational demand of the WLS optimization.

In a previous paper [M. Robbiano, E.A. Martins, and I. Gutman, Extending a theorem by Fiedler and applications to graph energy, MATCH Commun. Math. Comput. Chem. 64 (2010), pp. 145-156], a lemma by Fiedler was used to obtain eigenspaces of graphs, and applied to graph energy. In this article Fiedler's lemma is generalized and this generalization is applied to graph spectra and graph energy.

Due to the proliferation of drafting packages, there are a lot of vector drawings available for people to integrate into documents. Moreover, creative designers and drafts-people often reuse data from previous projects and libraries of ready-to-use components. Usually, retrieving these drawings is a slow, complex and error-prone endeavor. While text-driven attempts at classifying image data have been recently supplemented with query-by-image content, these work only for bitmap-type data and cannot handle vectorial information. Furthermore, they are not very good at retrieving partial information, which makes it difficult to search complex documents.

The Estrada index EE is a recently proposed molecular structure-descriptor, used in the modeling of certain features of the 3D structure of organic molecules, in particular of the degree of folding of proteins and other long-chain biopolymers. The Estrada index is computed from the spectrum of the molecular graph. Therefore, finding its relation with the spectral radius r (= the greatest graph eigenvalue) is of interest, especially because the structure-dependency of r is relatively well understood. In this work, the basic characteristics of the relation between EE and r, which turned out to be much more complicated than initially anticipated, was determined.

The Estrada index EE is a recently proposed molecular structure-descriptor, used in the modeling of certain features of the 3D structure of organic molecules, in particular of the degree of folding of proteins and other long-chain biopolymers. The Estrada index is computed from the spectrum of the molecular graph. Therefore, finding its relation with the spectral radius r (= the greatest graph eigenvalue) is of interest, especially because the structure-dependency of r is relatively well understood. In this work, the basic characteristics of the relation between EE and r, which turned out to be much more complicated than initially anticipated, was determined.

The Estrada index EE is a recently proposed molecular structure-descriptor, used in the modeling of certain features of the 3D structure of organic molecules, in particular of the degree of folding of proteins and other long-chain biopolymers. The Estrada index is computed from the spectrum of the molecular graph. Therefore, finding its relation with the spectral radius r (= the greatest graph eigenvalue) is of interest, especially because the structure-dependency of r is relatively well understood. In this work, the basic characteristics of the relation between EE and r, which turned out to be much more complicated than initially anticipated, was determined.

Let G be a graph on n vertices v1; v2; : : : ; vn and let d(vi) be the degree (= number of ÿrst neighbors) of the vertex vi. If (d(v1); d(v2); : : : ; d(vn)) t is an eigenvector of the (0; 1)-adjacency matrix of G, then G is said to be harmonic. Earlier all harmonic trees were determined; their number is inÿnite. We now show that for any c; c ¿ 1, the number of connected harmonic graphs with cyclomatic number c is ÿnite. In particular, there are no connected non-regular unicyclic and bicyclic harmonic graphs and there exist exactly four and eighteen connected non-regular tricyclic and tetracyclic harmonic graphs.

An increasing number of synthetic topology generators are available, each claiming to produce representative Internet topologies. Every generator has its own parameters, allowing the user to generate topologies with different characteristics. However, there exist no clear guidelines on tuning the value of these parameters in order to obtain a topology with specific characteristics. In this paper we optimize the parameters of several topology generators to match a given Internet topology. The optimization is performed either with respect to the link density, or to the spectrum of the normalized Laplacian matrix. Contrary to approaches in the literature that rely only on the largest eigenvalues, we take into account the set of all eigenvalues. However, we show that on their own the eigenvalues cannot be used to construct a metric for optimizing parameters. Instead we present a weighted spectral method which simultaneously takes into account all the properties of the graph.

The local spectrum of a graph G = (V, E), constituted by the standard eigenvalues of G and their local multiplicities, plays a similar role as the global spectrum when the graph is "seen" from a given vertex. Thus, for each vertex i ∈ V , the i-local multiplicities of all the eigenvalues add up to 1; whereas the multiplicity of each eigenvalue λ of G is the sum, extended to all vertices, of its local multiplicities. In this work, using the interpretation of an eigenvector as a charge distribution on the vertices, we compute the local spectrum of the line graph LG in terms of the local spectrum of the regular graph G it derives from. Furthermore, some applications of this result are derived as, for instance, some results about the number of circuits of LG.

The local spectrum of a graph G = (V, E), constituted by the standard eigenvalues of G and their local multiplicities, plays a similar role as the global spectrum when the graph is "seen" from a given vertex. Thus, for each vertex i ∈ V , the i-local multiplicities of all the eigenvalues add up to 1; whereas the multiplicity of each eigenvalue λ of G is the sum, extended to all vertices, of its local multiplicities. In this work, using the interpretation of an eigenvector as a charge distribution on the vertices, we compute the local spectrum of the line graph LG in terms of the local spectrum of the regular graph G it derives from. Furthermore, some applications of this result are derived as, for instance, some results about the number of circuits of LG.

The study of robustness and connectivity properties are important in the analysis of complex networks. This paper reports on an effort to compare different network topologies according to their algebraic connectivity, network criticality, average node degree, and average node betweenness. We consider different network types and study the behavior of these various metrics as scale is increased. Based on extensive simulations, we suggest some guidelines for the design and simplification of networks. The main finding is that, algebraic connectivity, network criticality, average degree, and average node betweenness capture different properties of a graph. Depending on the nature of the problem at hand, one needs to select which one is appropriate to use as the main metric for network analysis.

In this paper, we introduce a new measure of symmetry which we call the symmetry ratio of a network, defined to be the ratio of the number of distinct eigenvalues of the network to D+ 1, where D is the diameter. The symmetry ratio has utility in partially predicting the robustness of a network in the face of attack. We prove a number of results placing bounds on the symmetry ratio for several families of networks, including distance-transitive networks, prisms, twisted prisms, antiprisms, tori, Cayley graphs, and random graphs.

It is shown that only a fraction of 2 −Ω(n) of the graphs on n vertices have an integral spectrum. Although there are several explicit constructions of such graphs, no upper bound for their number has been known. Graphs of this type play an important role in quantum networks supporting the so-called perfect state transfer.

The energy of a graph G is the sum of the absolute values of the eigenvalues of the adjacency matrix of G. The Laplacian (respectively, the signless Laplacian) energy of G is the sum of the absolute values of the differences between the eigenvalues of the Laplacian (respectively, signless Laplacian) matrix and the arithmetic mean of the vertex degrees of the graph. In this paper, among some results which relate these energies, we point out some bounds to them using the energy of the line graph of G. Most of these bounds are valid for both energies, Laplacian and singless Laplacian. However, we present two new upper bounds on the signless Laplacian which are not upper bounds for the Laplacian energy.

Let G be a graph on n vertices, and let λ 1 , λ 2 , . . . , λ n be its eigenvalues. The Estrada index of G is a recently introduced graph invariant, defined as EE = n i=1 e λ i . We establish lower and upper bounds for EE in terms of the number of vertices and number of edges. Also some inequalities between EE and the energy of G are obtained.

We use the concept of network communicability (Phys. Rev. E 77, 036111 (2008)) to define communities in a complex network. The communities are defined as the cliques of a "communicability graph". The communicability graph is defined here as the graph having the same set of vertices as the complex network and the connections between them are determined by the communicability function. Two nodes are connected in this graph if their internal communicability is larger than the external communicability with the rest of nodes. We define that the connected nodes in the communicability graph are members of the same network community. Then, the problem of finding network communities is expressed in terms of the all-cliques problem. In addition, we extend here the concept of communicability to account for the strength of the interactions between the nodes. This "strength parameter" is identified as the inverse temperature of the network and can account for the spring constant of a spring network, the conductivity of a resister network or the bandwith of a telephone network. Furthermore, it can represent the level of stress which the node interactions receive, like the level of social agitation in a social network.

We investigate a certain condition for isomorphism between circulant graphs (known as the à Adà am property) in a stronger form by referring to isospectrality rather than to isomorphism of graphs. We describe a wide class of graphs for which the à Adà am conjecture holds. We apply these results to establish an asymptotic formula for the number of non-isomorphic circulant graphs and connected circulant graphs.