Spectral characterizations of signed lollipop graphs

Let Γ=(G,σ) be a signed graph, where G is the underlying simple graph and σ:E(G)→{+,-} is the sign function on the edges of G. In this paper we consider the spectral characterization problem extended to the adjacency matrix and Laplacian matrix of signed graphs. After giving some basic results, we study the spectral determination of signed lollipop graphs, and we show that any signed lollipop graph is determined by the spectrum of its Laplacian matrix

The spectral excess theorem for graphs with few eigenvalues whose distance- 2 or distance-1-or-2 graph is strongly regular

We study regular graphs whose distance-2 graph or distance-1-or-2 graph is strongly regular. We provide a characterization of such graphs Γ (among regular graphs with few distinct eigenvalues) in terms of the spectrum and the mean number of vertices at maximal distance d from every vertex, where d+1 is the number of different eigenvalues of Γ. This can be seen as another version of the so-called spectral excess theorem, which characterizes in a similar way those regular graphs that are distance-regular.Research of C. Dalfó and M. A. Fiol is partially supported by Agència de Gestió d'Ajuts Universitaris i de Recerca (AGAUR) under project 2017SGR1087. Research of J. Koolen is partially supported by the National Natural Science Foundation of China under project No. 11471009, and the Chinese Academy of Sciences under its ‘100 talent’ programme. The research of C. Dalfó has also received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 734922

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

Integral formulas with distribution kernels for irreducible projections in L2 of a nilmanifold

AbstractLet N be a simply connected nilpotent Lie group and Γ a discrete uniform subgroup. The authors consider irreducible representations σ in the spectrum of the quasi-regular representation N × L2(Γ/N) → L2(Γ→) which are induced from normal maximal subordinate subgroups M ⊆ N. The primary projection Pσ and all irreducible projections P ⩽ Pσ are given by convolutions involving right Γ-invariant distributions D on Γ→, Pf(Γn) = D ∗ f(Γn) = <D, n · f>all f ϵ C∞(Γ/N), where n · f(ζ) = f(ζ · n). Extending earlier work of Auslander and Brezin, and L. Richardson, the authors give explicit character formulas for the distributions, interpreting them as sums of characters on the torus Tκ = (Γ ∩ M) · [M, M]⧹M. By examining these structural formulas, they obtain fairly sharp estimates on the order of the distributions: if σ is associated with an orbit O ⊆ n∗ and if V ⊆ n∗ is the largest subspace which saturates θ in the sense that f ϵ O ⇒ f + V ⊆ O. As a corollary they obtain Richardson's criterion for a projection to map C0(Γ→) into itself. The authors also resolve a conjecture of Brezin, proving a Zero-One law which says, among other things, that if the primary projection Pσ maps Cr(Γ→) into C0(Γ→), so do all irreducible projections P ⩽ Pσ. This proof is based on a classical lemma on the extent to which integral points on a polynomial graph in Rn lie in the coset ring of Zn (the finitely additive Boolean algebra generated by cosets of subgroups in Zn). This lemma may be useful in other investigations of nilmanifolds

On Laplacian Eigenvalues of the Zero-Divisor Graph Associated to the Ring of Integers Modulo n

Given a commutative ring R with identity 1≠0, let the set Z(R) denote the set of zero-divisors and let Z*(R)=Z(R)∖0 be the set of non-zero zero-divisors of R. The zero-divisor graph of R, denoted by Γ(R), is a simple graph whose vertex set is Z*(R) and each pair of vertices in Z*(R) are adjacent when their product is 0. In this article, we find the structure and Laplacian spectrum of the zero-divisor graphs Γ(Zn) for n=pN1qN2, where p&lt;q are primes and N1,N2 are positive integers

From self-similar groups to self-similar sets and spectra

The survey presents developments in the theory of self-similar groups leading to applications to the study of fractal sets and graphs, and their associated spectra