On Spectral Integral Variations of Graphs

Let G be a general graph. The spectrum S ( G ) of G is defined to be the spectrum of its Laplacian matrix. Let G + e be the graph obtained from G by adding an edge or a loop e . We study in this paper when the spectral variation between G and G + e is integral and obtain some equivalent conditions, through which a new Laplacian integral graph can be constructed from a known Laplacian integral graph by adding an edge.     

On Spectral Integral Variations of Graphs

Let G be a general graph. The spectrum S ( G ) of G is defined to be the spectrum of its Laplacian matrix. Let G + e be the graph obtained from G by adding an edge or a loop e . We study in this paper when the spectral variation between G and G + e is integral and obtain some equivalent conditions, through which a new Laplacian integral graph can be constructed from a known Laplacian integral graph by adding an edge.     

On the Laplacian integral tricyclic graphs

A graph is called Laplacian integral if all its Laplacian eigenvalues are integers. In this paper, we give an edge subdividing theorem for Laplacian eigenvalues of a graph (Theorem 2.1) and characterize a class of k-cyclic graphs whose algebraic connectivity is less than one. Using these results, we determine all the Laplacian integral tricyclic graphs. Furthermore, we show that all the Laplacian integral tricyclic graphs are determined by their Laplacian spectra.     

Laplacian Spectrum of Weakly Quasi-threshold Graphs

In this paper we study the class of weakly quasi-threshold graphs that are obtained from a vertex by recursively applying the operations (i) adding a new isolated vertex, (ii) adding a new vertex and making it adjacent to all old vertices, (iii) disjoint union of two old graphs, and (iv) adding a new vertex and making it adjacent to all neighbours of an old vertex. This class contains the class of quasi-threshold graphs. We show that weakly quasi-threshold graphs are precisely the comparability graphs of a forest consisting of rooted trees with each vertex of a tree being replaced by an independent set. We also supply a quadratic time algorithm in the the size of the vertex set for recognizing such a graph. We completely determine the Laplacian spectrum of weakly quasi-threshold graphs. It turns out that weakly quasi-threshold graphs are Laplacian integral. As a corollary we obtain a closed formula for the number of spanning trees in such graphs. A conjecture of Grone and Merris asserts that the spectrum of the Laplacian of any graph is majorized by the conjugate of the degree sequence of the graph. We show that the conjecture holds for cographs. Prof. Bapat and Prof. Pati take this opportunity to thank the Indian Institute of Technology Kanpur for the invitation. The authors also wish to thank the Department of Science and Technology, New Delhi for the project grant.     

Spectral Integral Variations of Degree Maximal Graphs

The concept of the spectral integral variation was introduced by Fan [Fan Yizheng (2002). On spectral integral variations of graphs. Linear and Multilinear Algebra , 50 , 133-142] to study the general graphs with all changed eigenvalues moving up by integers when an edge is added. Here we consider the spectral integral variations of maximal graphs G , and successfully give an equivalent condition for the spectral integral variation of G occurring in two places by adding an edge e . We also characterize whether the graph G + e is maximal so that an explicit interpretation of the above condition is obtained, where G + e denotes the graph obtained from G by adding an edge e .     

Spectral Integral Variations of Degree Maximal Graphs

The concept of the spectral integral variation was introduced by Fan [Fan Yizheng (2002). On spectral integral variations of graphs. Linear and Multilinear Algebra , 50 , 133-142] to study the general graphs with all changed eigenvalues moving up by integers when an edge is added. Here we consider the spectral integral variations of maximal graphs G , and successfully give an equivalent condition for the spectral integral variation of G occurring in two places by adding an edge e . We also characterize whether the graph G + e is maximal so that an explicit interpretation of the above condition is obtained, where G + e denotes the graph obtained from G by adding an edge e .     

A characterization of spectral integral variation in two places for Laplacian matrices

We describe the graphs having the property that adding a particular edge will result in exactly two Laplacian eigenvalues increasing by 1 and the other Laplacian eigenvalues remaining fixed. For a certain subclass of graphs, we also characterize the Laplacian integral graphs with that property. Finally, we investigate a situation in which the algebraic connectivity is one of the eigenvalues that increases by 1.     

Indecomposable laplacian integral graphs

A graph that can be constructed from isolated vertices by the operations of union and complement is decomposable. Every decomposable graph is Laplacian integral. i.e., its Laplacian spectrum consists entirely of integers. An indecomposable graph is not decomposable. The main purpose of this note is to demonstrate the existence of infinitely many indecomposable Laplacian integral graphs.     

Half Sampling on Bipartite Graphs

On a bipartite graph G we consider the half sampling problem of uniquely recovering a function from its values on the even vertices, under the appropriate half bandlimited assumption with respect to a Laplacian on the graph. We discuss both finite and infinite graphs, give the appropriate definition of “half bandlimited” that involves splitting the mid frequency, and give an explicit solution to the problem. We discuss in detail the example of a regular tree. We also consider a related sampling problem on graphs that are generated by edge substitution.     

Commuting projections on graphs

Motivated by the increasing importance of large‐scale networks typically modeled by graphs, this paper is concerned with the development of mathematical tools for solving problems associated with the popular graph Laplacian. We exploit its mixed formulation based on its natural factorization as product of two operators. The goal is to construct a coarse version of the mixed graph Laplacian operator with the purpose to construct two‐level, and by recursion, a multilevel hierarchy of graphs and associated operators. In many situations in practice, having a coarse (i.e., reduced dimension) model that maintains some inherent features of the original large‐scale graph and respective graph Laplacian offers potential to develop efficient algorithms to analyze the underlined network modeled by this large‐scale graph. One possible application of such a hierarchy is to develop multilevel methods that have the potential to be of optimal complexity. In this paper, we consider general (connected) graphs and function spaces defined on its edges and its vertices. These two spaces are related by a discrete gradient operator, ‘Grad’ and its adjoint, ‘ ? Div’, referred to as (negative) discrete divergence. We also consider a coarse graph obtained by aggregation of vertices of the original one. Then, a coarse vertex space is identified with the subspace of piecewise constant functions over the aggregates. We consider the ?₂‐projection Q_H onto the space of these piecewise constants. In the present paper, our main result is the construction of a projection π_H from the original edge‐space onto a properly constructed coarse edge‐space associated with the edges of the coarse graph. The projections π_H and Q_H commute with the discrete divergence operator, that is, we have Div π_H = Q_H div. The respective pair of coarse edge‐space and coarse vertex‐space offer the potential to construct two‐level, and by recursion, multilevel methods for the mixed formulation of the graph Laplacian, which utilizes the discrete divergence operator. The performance of one two‐level method with overlapping Schwarz smoothing and correction based on the constructed coarse spaces for solving such mixed graph Laplacian systems is illustrated on a number of graph examples. Copyright © 2013 John Wiley & Sons, Ltd.     

双圈图的Laplace谱跨度

The Laplacian spread of a graph is defined to be the difference between the largest eigenvalue and the second smallest eigenvalue of the Laplacian matrix of the graph. In our recent work, we have determined the graphs with maximal Laplacian spreads among all trees of fixed order and among all unicyclic graphs of fixed order, respectively. In this paper, we continue the work on Laplacian spread of graphs, and prove that there exist exactly two bicyclic graphs with maximal Laplacian spread among all bicyclic graphs of fixed order, which are obtained from a star by adding two incident edges and by adding two nonincident edges between the pendant vertices of the star, respectively.     

Braess's paradox for the spectral gap in random graphs and delocalization of eigenvectors

We study how the spectral gap of the normalized Laplacian of a random graph changes when an edge is added to or removed from the graph. There are known examples of graphs where, perhaps counter‐intuitively, adding an edge can decrease the spectral gap, a phenomenon that is analogous to Braess's paradox in traffic networks. We show that this is often the case in random graphs in a strong sense. More precisely, we show that for typical instances of Erd?s‐Rényi random graphs G (n, p ) with constant edge density , the addition of a random edge will decrease the spectral gap with positive probability, strictly bounded away from zero. To do this, we prove a new delocalization result for eigenvectors of the Laplacian of G (n, p ), which might be of independent interest. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 584–611, 2017     

The eigenvalues of the Laplacian on locally finite networks

We consider the continuous Laplacian on an infinite locally finite network with equal edge lengths under natural transition conditions as continuity at the ramification nodes and classical Kirchhoff conditions at all vertices. It is shown that eigenvalues of the Laplacian in a L^∞-setting are closely related to those of the adjacency and transition operator of the network. In this way the point spectrum is determined completely in terms of combinatorial quantities and properties of the underlying graph as in the finite case [2]. Moreover, the occurrence of infinite geometric multiplicity on trees and some periodic graphs is investigated.     

On the Laplacian Eigenvalues of Signed Graphs

A signed graph is a graph with a sign attached to each edge. This article extends some fundamental concepts of the Laplacian matrices from graphs to signed graphs. In particular, the largest Laplacian eigenvalue of a signed graph is investigated, which generalizes the corresponding results on the largest Laplacian eigenvalue of a graph.     

Q‐integral unicyclic,bicyclic and tricyclic graphs

A graph is called Q‐integral if its signless Laplacian spectrum consists of integers. In this paper, we characterize a class of k‐cyclic graphs whose second smallest signless Laplacian eigenvalue is less than one. Using this result we determine all the Q‐integral unicyclic, bicyclic and tricyclic graphs.     

Connected Q-integral graphs with maximum edge-degree less than or equal to 8

Graphs with integral signless Laplacian spectrum are called Q-integral graphs. The number of adjacent edges to an edge is defined as the edge-degree of that edge. The Q-spectral radius of a graph is the largest eigenvalue of its signless Laplacian. In 2019, Park and Sano [16] studied connected Q-integral graphs with the maximum edge-degree at most six. In this article, we extend their result and study the connected Q-integral graphs with maximum edge-degree less than or equal to eight. Further, we give an upper bound and a lower bound for the maximum edge-degree of a connected Q-integral graph with respect to its Q-spectral radius. As a corollary, we show that the Q-spectral radius of the connected edge-non-regular Q-integral graph with maximum edge-degree five is six, which we anticipate to be a key for solving the unsolved problem of characterizing such graphs.     

The Laplacian spectra of graphs with a tree structure

Trees are very common in the theory and applications of combinatorics. In this article, we consider graphs whose underlying structure is a tree, except that its vertices are graphs in their own right and where adjacent graphs (vertices) are linked by taking their join. We study the spectral properties of the Laplacian matrices of such graphs. It turns out that in order to capture known spectral properties of the Laplacian matrices of trees, it is necessary to consider the Laplacians of vertex-weighted graphs. We focus on the second smallest eigenvalue of such Laplacians and on the properties of their corresponding eigenvector. We characterize the second smallest eigenvalue in terms of the Perron branches of a tree. Finally, we show that our results are applicable to advancing the solution to the problem of whether there exists a graph on n vertices whose Laplacian has the integer eigenvalues 0, 1, …, n ? 1.     

Equality of Lifshitz and van Hove exponents on amenable Cayley graphs

We study the low energy asymptotics of periodic and random Laplace operators on Cayley graphs of amenable, finitely generated groups. For the periodic operator the asymptotics is characterised by the van Hove exponent or zeroth Novikov–Shubin invariant. The random model we consider is given in terms of an adjacency Laplacian on site or edge percolation subgraphs of the Cayley graph. The asymptotic behaviour of the spectral distribution is exponential, characterised by the Lifshitz exponent. We show that for the adjacency Laplacian the two invariants/exponents coincide. The result holds also for more general symmetric transition operators. For combinatorial Laplacians one has a different universal behaviour of the low energy asymptotics of the spectral distribution function, which can be actually established on quasi-transitive graphs without an amenability assumption. The latter result holds also for long range bond percolation models.     

The (signless) Laplacian spectral radius of unicyclic and bicyclic graphs with <Emphasis Type="Italic">n</Emphasis> vertices and <Emphasis Type="Italic">k</Emphasis> pendant vertices

In this paper, the effects on the signless Laplacian spectral radius of a graph are studied when some operations, such as edge moving, edge subdividing, are applied to the graph. Moreover, the largest signless Laplacian spectral radius among the all unicyclic graphs with n vertices and k pendant vertices is identified. Furthermore, we determine the graphs with the largest Laplacian spectral radii among the all unicyclic graphs and bicyclic graphs with n vertices and k pendant vertices, respectively.