Eigenvalues Of Graph Research Articles

Improved Lower Bounds on the Extrema of Eigenvalues of Graphs

Improved Lower Bounds on the Extrema of Eigenvalues of Graphs

Sharp bounds on the least eigenvalue of a graph determined from edge clique partitions

Sharp bounds on the least eigenvalue of an arbitrary graph are presented. Necessary and sufficient (just sufficient) conditions for the lower (upper) bound to be attained are deduced using edge clique partitions. As an application, we prove that the least eigenvalue of the n-Queens graph {mathcal {Q}}(n) is equal to -4 for every n ge 4 and it is also proven that the multiplicity of this eigenvalue is (n-3)^2. Finally, edge clique partitions of additional infinite families of connected graphs and their relations with the least eigenvalues are presented.

No Laplacian perfect state transfer in total graphs

The total graph of a graph G, denoted by T(G), is defined to be the graph whose vertices are the vertices and edges of G, with two vertices of T(G) adjacent if and only if the corresponding elements of G are adjacent or incident. In [Discrete Mathematics, 344 (2021) 112139], Liu and Wang proved that if r+1 is not a Laplacian eigenvalue of an r-regular graph G, then there is no Laplacian perfect state transfer in T(G). They also posed the question: If r+1 is a Laplacian eigenvalue of an r-regular graph G, can T(G) have LPST? In this paper, we completely answer this question. We give new descriptions concerning the eigenvalue support and spectral decomposition of the Laplacian matrix of T(G), by use of which we show that there is in fact no Laplacian perfect state transfer in T(G) for any regular graph G.

On Graphs with c2-c3 Successive Minimal Laplacian Coefficients

Let G be a graph of order n and L(G) be its Laplacian matrix. The Laplacian polynomial of G is defined as P(G;λ)=det(λI−L(G))=∑i=0n(−1)ici(G)λn−i, where ci(G) is called the i-th Laplacian coefficient of G. Denoted by Gn,m the set of all (n,m)-graphs, in which each of them contains n vertices and m edges. The graph G is called uniformly minimal if, for each i(i=0,1,…,n), H is ci(G)-minimal in Gn,m. The Laplacian matrix and eigenvalues of graphs have numerous applications in various interdisciplinary fields, such as chemistry and physics. Specifically, these matrices and eigenvalues are widely utilized to calculate the energy of molecular energy and analyze the physical properties of materials. The Laplacian-like energy shares a number of properties with the usual graph energy. In this paper, we investigate the existence of uniformly minimal graphs in Gn,m because such graphs have minimal Laplacian-like energy. We determine that the c2(G)-c3(G) successive minimal graph is exactly one of the four classes of threshold graphs.

Enumeration of Perfect Matchings of the Cartesian Products of Graphs

A subgraph $ H $ of a graph $G$ is nice if $ G-V(H) $ has a perfect matching. An even cycle $ C $ in an oriented graph is oddly oriented if for either choice of direction of traversal around $ C $, the number of edges of $C$ directed along the traversal is odd. An orientation $ D $ of a graph $ G $ with an even number of vertices is Pfaffian if every nice cycle of $ G $ is oddly oriented in $ D $. Let $ P_{n} $ denote a path on $ n $ vertices. The Pfaffian graph $G \times P_{2n} $ was determined by Lu and Zhang [The Pfaffian property of Cartesian products of graphs, J. Comb. Optim. 27 (2014) 530--540]. In this paper, we characterize the Pfaffian graph $ G \times P_{2n+1} $ with respect to the forbidden subgraphs of $G$. We first give sufficient and necessary conditions under which $G\times P_{2n+1}$ ($n\geqslant 2$) is Pfaffian. Then we characterize the Pfaffian graph $ G \times P_{3} $ when $G$ is a bipartite graph, and we generalize this result to the the case $G$ contains exactly one odd cycle. Following these results, we enumerate the number of perfect matchings of the Pfaffian graph $G \times P_{n}$ in terms of the eigenvalues of the orientation graph of $G$, and we also count perfect matchings of some Pfaffian graph $G \times P_{n}$ by the eigenvalues of $G$.

ON SPECTRA OF POWER GRAPHS OF FINITE CYCLIC AND DIHEDRAL GROUPS

The power graph 𝒫(G) of a finite group G is defined to be the graph whose vertex set is G and in which two distinct vertices u,v∈𝒫(G) are adjacent if and only if u=vm or v=un for some positive integers m,n. The distance signless Laplacian matrix of a graph 𝒢, denoted by DQ(𝒢), is defined as DQ(𝒢)=Tr(𝒢)+D(𝒢), where D(𝒢) is the distance matrix of 𝒢 and Tr(𝒢) is the transmission matrix of 𝒢. We determine the distance signless Laplacian eigenvalues of the power graphs of the finite cyclic group ℤn and the dihedral group Dn. We provide upper and lower bounds on the largest eigenvalue of the distance signless Laplacian matrix of 𝒫(ℤn) and 𝒫(Dn). We also give a short proof of the lower bound on the algebraic connectivity of 𝒫(ℤn) obtained by Chattopadhyay and Panigrahi (Linear and Multilinear Algebra 63:7 (2015), 1345–1355).

On irregularity indices and main eigenvalues of graphs and their applicability

On irregularity indices and main eigenvalues of graphs and their applicability

Energy of Graph

By given the adjacency matrix, laplacian matrix of a graph we can find the set of eigenvalues of graph in order to discussed about the energy of graph and laplacian energy of graph. (i.e. the sum of eigenvalues of adjacency matrix and laplacian matrix of a graph is called the energy of graph) and the laplacian energy of graph is greater or equal to zero for any graph and is greater than zero for every connected graph with more or two vertices (i.e. the last eigenvalues of laplacian matrix is zero), according to several theorems about the energy of graph and the laplacian energy of graph that are described in this work; I discussed about energy of graph, laplacian energy of graph and comparing them here.

Distance Laplacian spectral ordering of sun type graphs

Let G be a simple, connected graph of order n. Its distance Laplacian energy DLE(G) is given by DLE(G)=∑i=1n|ρiL−2W(G)n|, where ρ1L≥ρ2L≥⋯≥ρnL are the distance Laplacian eigenvalues and W(G) is the Wiener index of G. Distance Laplacian eigenvalues of sun and partial sun graphs have been characterized. We order the partial sun graphs by using their second largest distance Laplacian eigenvalue. Moreover, the distance Laplacian energy of sun and partial sun graphs have been derived in this paper. These graphs are also ordered by using their distance Laplacian energies.

Signed spectral Turań type theorems

A signed graph Σ=(G,σ) is a graph where the function σ assigns either 1 or −1 to each edge of the simple graph G. The adjacency matrix of Σ, denoted by A(Σ), is defined canonically. In a recent paper, Wang et al. extended the spectral bounds of Hoffman and Cvetković for the chromatic number of signed graphs. They proposed an open problem related to the balanced clique number and the largest eigenvalue of a signed graph. We solve a strengthened version of this open problem. As a byproduct, we give alternate proofs for some of the known classical bounds for the least eigenvalues of unsigned graphs. We extend the Turán's inequality for the signed graphs. Besides, we study the Bollobás and Nikiforov conjecture for the signed graphs and show that the conjecture need not be true for the signed graphs. Nevertheless, the conjecture holds for signed graphs under some assumptions. Finally, we study some of the relationships between the number of signed walks and the largest eigenvalue of a signed graph.

Remarks on the sum of powers of normalized signless Laplacian eigenvalues of graphs

Let G = (V, E), V = {v1, v2,..., vn}, be a simple connected graph of order n and size m. Denote by ?+1 ? ?+2 ?...? ?+n ? 0 the normalized signless Laplacian eigenvalues of G, and by ??(G) the sum of ?-th powers of the normalized signless Laplacian eigenvalues of a connected graph. The paper deals with bounds of ??. Some special cases, when ? = 12 and ? = ?1, are also considered.

Remaining popular: power-law regularities in network dynamics

The structure of networks has been a focal research topic over the past few decades. These research efforts have enabled the discovery of numerous structural patterns and regularities, bringing forth advancements in many fields. In particular, the ubiquitous power-law patterns evident in degree distributions, graph eigenvalues and human mobility patterns have provided the opportunity to model many different complex systems. However, regularities in the dynamical patterns of networks remain a considerably less explored terrain. In this study we examine the dynamics of networks, focusing on stability characteristics of node popularity, and present our results using various empirical datasets. Specifically, we address several intriguing questions – for how long are popular nodes expected to remain so? How much time is expected to pass between two consecutive popularity periods? What characterizes nodes which manage to maintain their popularity for long periods of time? Surprisingly, we find that such temporal aspects are governed by a power-law regime, and that these power-law regularities are equally likely across all node ages.

A bound for the p-domination number of a graph in terms of its eigenvalue multiplicities

Let G be a connected graph of order n with domination number γ(G). Wang, Yan, Fang, Geng and Tian [Linear Algebra Appl. 607 (2020), 307-318] showed that for any Laplacian eigenvalue λ of G with multiplicity mG(λ), it holds that γ(G)≤n−mG(λ). Using techniques from the theory of star sets, in this work we prove that the same bound holds when λ is an arbitrary adjacency eigenvalue of a non-regular graph, and we characterize the cases of equality. Moreover, we show a result that gives a relationship between start sets and the p-domination number, and we apply it to extend the aforementioned spectral bound to the p-domination number using the adjacency and Laplacian eigenvalue multiplicities.

Alternating sign property of the perfect matching derangement graph

It was conjectured in the monograph [9] by Godsil and Meagher and in the article [10] by Lindzey that the perfect matching derangement graph M2n possesses the alternating sign property, that is, for any integer partition λ=(λ1,…,λr)⊢n, the sign of the eigenvalue ηλ of M2n is given by sign(ηλ)=(−1)n−λ1. In this paper, we prove that the conjecture is true. Our approach yields a recurrence formula for the eigenvalues of the perfect matching derangement graph as well as a new recurrence formula for the eigenvalues of the permutation derangement graph.

The Maximum Spectral Radius of Non-Bipartite Graphs Forbidding Short Odd Cycles

It is well-known that eigenvalues of graphs can be used to describe structural properties and parameters of graphs. A theorem of Nosal and Nikiforov states that if $G$ is a triangle-free graph with $m$ edges, then $\lambda (G)\le \sqrt{m}$, equality holds if and only if $G$ is a complete bipartite graph. Recently, Lin, Ning and Wu [Combin. Probab. Comput. 30 (2021)] proved a generalization for non-bipartite triangle-free graphs. Moreover, Zhai and Shu [Discrete Math. 345 (2022)] presented a further improvement. In this paper, we present an alternative method for proving the improvement by Zhai and Shu. Furthermore, the method can allow us to give a refinement on the result of Zhai and Shu for non-bipartite graphs without short odd cycles.

Spektrum Laplace pada graf kincir angin berarah (Q_k^3)

Suppose that 0 = µ0 ≤ µ1 ≤ ... ≤ µn-1 are eigen values of a Laplacian matrix graph with n vertices and m(µ0), m(µ1), …, m(µn-1) are the multiplicity of each µ, so the Laplacian spectrum of a graph can be expressed as a matrix 2 × n whose line elements are µ0, µ1, …, µn-1 for the first row, and m(µ0), m(µ1), …, m(µn-1) for the second row. In this paper, we will discuss Laplacian spectrum of the directed windmill graph () with k ≥ 1. The determination of the Laplacian spectrum in this study is to determine the characteristic polynomial of the Laplacian matrix from the directed windmill graph () with k ≥ 1.
 Keywords: Characteristic polynomial, directed windmill graph, Laplacian matrix, Laplacian spectrum.MSC2020 :05C50

On the Sum and Spread of Reciprocal Distance Laplacian Eigenvalues of Graphs in Terms of Harary Index

The reciprocal distance Laplacian matrix of a connected graph G is defined as RDL(G)=RT(G)−RD(G), where RT(G) is the diagonal matrix of reciprocal distance degrees and RD(G) is the Harary matrix. Clearly, RDL(G) is a real symmetric matrix, and we denote its eigenvalues as λ1(RDL(G))≥λ2(RDL(G))≥…≥λn(RDL(G)). The largest eigenvalue λ1(RDL(G)) of RDL(G), denoted by λ(G), is called the reciprocal distance Laplacian spectral radius. In this paper, we obtain several upper bounds for the sum of k largest reciprocal distance Laplacian eigenvalues of G in terms of various graph parameters, such as order n, maximum reciprocal distance degree RTmax, minimum reciprocal distance degree RTmin, and Harary index H(G) of G. We determine the extremal cases corresponding to these bounds. As a consequence, we obtain the upper bounds for reciprocal distance Laplacian spectral radius λ(G) in terms of the parameters as mentioned above and characterize the extremal cases. Moreover, we attain several upper and lower bounds for reciprocal distance Laplacian spread RDLS(G)=λ1(RDL(G))−λn−1(RDL(G)) in terms of various graph parameters. We determine the extremal graphs in many cases.

On weighted spectral radius of unraveled balls and normalized Laplacian eigenvalues

For a graph G, the unraveled ball of radius r centered at a vertex v is the ball of radius r centered at v in the universal cover of G. We obtain a lower bound on the weighted spectral radius of unraveled balls of fixed radius in a graph with positive weights on edges, which is used to present an upper bound on the sth (where s≥2) smallest normalized Laplacian eigenvalue of irregular graphs under minor assumptions. Moreover, when s=2, the result may be regarded as an Alon–Boppana type bound for a class of irregular graphs.

Periodicity of Grover walks on bipartite regular graphs with at most five distinct eigenvalues

We determine connected bipartite regular graphs with four distinct adjacency eigenvalues that induce periodic Grover walks, and show that it is only C6. We also show that there are only three kinds of the second largest eigenvalues of bipartite regular periodic graphs with five distinct eigenvalues. Using walk-regularity, we enumerate feasible spectra for such graphs.

$$\ell $$-Connectivity, Integrity, Tenacity, Toughness and Eigenvalues of Graphs

$$\ell $$-Connectivity, Integrity, Tenacity, Toughness and Eigenvalues of Graphs