Spectral characterizations of sun graphs and broken sun graphs

Graphs and AlgorithmsInternational audienceSeveral matrices can be associated to a graph such as the adjacency matrix or the Laplacian matrix. The spectrum of these matrices gives some informations about the structure of the graph and the question ''Which graphs are determined by their spectrum?'' remains a difficult problem in algebraic graph theory. In this article we enlarge the known families of graphs determined by their spectrum by considering some unicyclic graphs. An odd (resp. even) sun is a graph obtained by appending a pendant vertex to each vertex of an odd (resp. even) cycle. A broken sun is a graph obtained by deleting pendant vertices of a sun. In this paper we prove that a sun is determined by its Laplacian spectrum, an odd sun is determined by its adjacency spectrum (counter-examples are given for even suns) and we give some spectral characterizations of broken suns

Laplacian spectral characterization of some double starlike trees

A tree is called double starlike if it has exactly two vertices of degree greater than two. Let $H(p,n,q)$ denote the double starlike tree obtained by attaching $p$ pendant vertices to one pendant vertex of the path $P_n$ and $q$ pendant vertices to the other pendant vertex of $P_n$. In this paper, we prove that $H(p,n,q)$ is determined by its Laplacian spectrum

Spectral characterizations of propeller graphs

A propeller graph is obtained from an $\infty$-graph by attaching a path to the vertex of degree four, where an $\infty$-graph consists of two cycles with precisely one common vertex. In this paper, we prove that all propeller graphs are determined by their Laplacian spectra as well as their signless Laplacian spectra

Laplacian Spectral Characterization of Signed Sun Graphs

A sun $SG_{n}$ is a graph of order $2n$ consisting of a cycle $C_{n}$, $n\geq 3$, to each vertex of it a pendant edge is attached. In this paper, we prove that unbalanced signed sun graphs are determined by their Laplacian spectra. Also we show that a balanced signed sun graph is determined by its Laplacian spectrum if and only if $n$ is odd

Graphs determined by their generalized characteristic polynomials

AbstractFor a given graph G with (0,1)-adjacency matrix AG, the generalized characteristic polynomial of G is defined to be ϕG=ϕG(λ,t)=det(λI-(AG-tDG)), where I is the identity matrix and DG is the diagonal degree matrix of G. In this paper, we are mainly concerned with the problem of characterizing a given graph G by its generalized characteristic polynomial ϕG. We show that graphs with the same generalized characteristic polynomials have the same degree sequence, based on which, a unified approach is proposed to show that some families of graphs are characterized by ϕG. We also provide a method for constructing graphs with the same generalized characteristic polynomial, by using GM-switching

Laplacian spectral properties of signed circular caterpillars

A circular caterpillar of girth n is a graph such that the removal of all pendant vertices yields a cycle Cn of order n. A signed graph is a pair Γ = (G, σ), where G is a simple graph and σ ∶ E(G) → {+1, −1} is the sign function defined on the set E(G) of edges of G. The signed graph Γ is said to be balanced if the number of negatively signed edges in each cycle is even, and it is said to be unbalanced otherwise. We determine some bounds for the first n Laplacian eigenvalues of any signed circular caterpillar. As an application, we prove that each signed spiked triangle (G(3; p, q, r), σ), i. e. a signed circular caterpillar of girth 3 and degree sequence πp,q,r = (p + 2, q + 2, r + 2, 1,..., 1), is determined by its Laplacian spectrum up to switching isomorphism. Moreover, in the set of signed spiked triangles of order N, we identify the extremal graphs with respect to the Laplacian spectral radius and the first two Zagreb indices. It turns out that the unbalanced spiked triangle with degree sequence πN−3,0,0 and the balanced spike triangle (G(3; p, ^ q, ^ r^), +), where each pair in {p, ^ q, ^ r^} differs at most by 1, respectively maximizes and minimizes the Laplacian spectral radius and both the Zagreb indices

Laplacian Spectral Characterization of Some Unicyclic Graphs

Let W(n;q,m1,m2) be the unicyclic graph with n vertices obtained by attaching two paths of lengths m1 and m2 at two adjacent vertices of cycle Cq. Let U(n;q,m1,m2,…,ms) be the unicyclic graph with n vertices obtained by attaching s paths of lengths m1,m2,…,ms at the same vertex of cycle Cq. In this paper, we prove that W(n;q,m1,m2) and U(n;q,m1,m2,…,ms) are determined by their Laplacian spectra when q is even

Discovering important nodes of complex networks based on laplacian spectra

© 2023 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes,creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.Knowledge of the Laplacian eigenvalues of a network provides important insights into its structural features and dynamical behaviours. Node or link removal caused by possible outage events, such as mechanical and electrical failures or malicious attacks, significantly impacts the Laplacian spectra. This can also happen due to intentional node removal against which, increasing the algebraic connectivity is desired. In this article, an analytical metric is proposed to measure the effect of node removal on the Laplacian eigenvalues of the network. The metric is formulated based on the local multiplicity of each eigenvalue at each node, so that the effect of node removal on any particular eigenvalues can be approximated using only one single eigen-decomposition of the Laplacian matrix. The metric is applicable to undirected networks as well as strongly-connected directed ones. It also provides a reliable approximation for the “Laplacian energy” of a network. The performance of the metric is evaluated for several synthetic networks and also the American Western States power grid. Results show that this metric has a nearly perfect precision in correctly predicting the most central nodes, and significantly outperforms other comparable heuristic methods.This research was partly supported by the Erasmus+ KA107 grant. AMA, MJ, LS and XY were supported by the Australian Research Council through project No. DP170102303. MJ and XY are also supported by the Australian Research Council through project No. DP200101199. MAF was supported by AGAUR from the Catalan Government under project 2017SGR1087, and by MICINN from the Spanish Government with the European Regional Development Fund under project PGC2018-095471-B-I00Peer ReviewedPostprint (author's final draft