On graphs whose second least eigenvalue is at least $-1$

M. Petrović

Displaying similar documents to “On graphs whose second least eigenvalue is at least $- 1$ ”

Behaviour of the first eigenvalue of the p-Laplacian in a domain with a hole

M. Sango (2001)

Colloquium Mathematicae

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We investigate the behaviour of a sequence $λ_{s}$ , s = 1,2,..., of eigenvalues of the Dirichlet problem for the p-Laplacian in the domains $Ω_{s}$ , s = 1,2,..., obtained by removing from a given domain Ω a set $E_{s}$ whose diameter vanishes when s → ∞. We estimate the deviation of $λ_{s}$ from the eigenvalue of the limit problem. For the derivation of our results we construct an appropriate asymptotic expansion for the sequence of solutions of the original eigenvalue problem.

Some properties complementary to Brualdi-Li matrices

Chuanlong Wang, Xuerong Yong (2015)

Czechoslovak Mathematical Journal

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In this paper we derive new properties complementary to an $2 n \times 2 n$ Brualdi-Li tournament matrix $B_{2 n}$ . We show that $B_{2 n}$ has exactly one positive real eigenvalue and one negative real eigenvalue and, as a by-product, reprove that every Brualdi-Li matrix has distinct eigenvalues. We then bound the partial sums of the real parts and the imaginary parts of its eigenvalues. The inverse of $B_{2 n}$ is also determined. Related results obtained in previous articles are proven to be corollaries.

Remarks on $D$ -integral complete multipartite graphs

Pavel Híc, Milan Pokorný (2016)

Czechoslovak Mathematical Journal

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A graph is called distance integral (or $D$ -integral) if all eigenvalues of its distance matrix are integers. In their study of $D$ -integral complete multipartite graphs, Yang and Wang (2015) posed two questions on the existence of such graphs. We resolve these questions and present some further results on $D$ -integral complete multipartite graphs. We give the first known distance integral complete multipartite graphs $K_{p_{1}, p_{2}, p_{3}}$ with $p_{1} < p_{2} < p_{3}$ , and $K_{p_{1}, p_{2}, p_{3}, p_{4}}$ with $p_{1} < p_{2} < p_{3} < p_{4}$ , as well as the infinite classes of distance integral...

Equivalent classes for K₃-gluings of wheels

Halina Bielak (1998)

Discussiones Mathematicae Graph Theory

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In this paper, the chromaticity of K₃-gluings of two wheels is studied. For each even integer n ≥ 6 and each odd integer 3 ≤ q ≤ [n/2] all K₃-gluings of wheels $W_{q + 2}$ and $W_{n - q + 2}$ create an χ-equivalent class.

Intrinsic linking and knotting are arbitrarily complex

Erica Flapan, Blake Mellor, Ramin Naimi (2008)

Fundamenta Mathematicae

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We show that, given any n and α, any embedding of any sufficiently large complete graph in ℝ³ contains an oriented link with components Q₁, ..., Qₙ such that for every i ≠ j, $| l k (Q_{i}, Q_{j}) | \geq α$ and $| a ₂ (Q_{i}) | \geq α$ , where $a ₂ (Q_{i})$ denotes the second coefficient of the Conway polynomial of $Q_{i}$ .

Global continuum of positive solutions for discrete $p$ -Laplacian eigenvalue problems

Dingyong Bai, Yuming Chen (2015)

Applications of Mathematics

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We discuss the discrete $p$ -Laplacian eigenvalue problem, $\{\begin{matrix} Δ (φ_{p} (Δ u (k - 1))) + λ a (k) g (u (k)) = 0, k \in {1, 2, ..., T}, \\ u (0) = u (T + 1) = 0, \end{matrix}$ where $T > 1$ is a given positive integer and $φ_{p} (x) : = {| x |}^{p - 2} x$ , $p > 1$ . First, the existence of an unbounded continuum $𝒞$ of positive solutions emanating from $(λ, u) = (0, 0)$ is shown under suitable conditions on the nonlinearity. Then, under an additional condition, it is shown that the positive solution is unique for any $λ > 0$ and all solutions are ordered. Thus the continuum $𝒞$ is a monotone continuous curve globally defined for all $λ > 0$ .

Estimates of the principal eigenvalue of the $p$ -Laplacian and the $p$ -biharmonic operator

Jiří Benedikt (2015)

Mathematica Bohemica

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We survey recent results concerning estimates of the principal eigenvalue of the Dirichlet $p$ -Laplacian and the Navier $p$ -biharmonic operator on a ball of radius $R$ in $ℝ^{N}$ and its asymptotics for $p$ approaching $1$ and $\infty$ . Let $p$ tend to $\infty$ . There is a critical radius $R_{C}$ of the ball such that the principal eigenvalue goes to $\infty$ for $0 < R \leq R_{C}$ and to $0$ for $R > R_{C}$ . The critical radius is $R_{C} = 1$ for any $N \in ℕ$ for the $p$ -Laplacian and $R_{C} = \sqrt{2 N}$ in the case of the $p$ -biharmonic operator. When $p$ approaches $1$ , the principal eigenvalue...

On the spectrum of the operator which is a composition of integration and substitution

Ignat Domanov (2008)

Studia Mathematica

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Let ϕ: [0,1] → [0,1] be a nondecreasing continuous function such that ϕ(x) > x for all x ∈ (0,1). Let the operator $V_{ϕ} : f (x) \mapsto \int_{0}^{ϕ (x)} f (t) d t$ be defined on L₂[0,1]. We prove that $V_{ϕ}$ has a finite number of nonzero eigenvalues if and only if ϕ(0) > 0 and ϕ(1-ε) = 1 for some 0 < ε < 1. Also, we show that the spectral trace of the operator $V_{ϕ}$ always equals 1.

On characterization of uniquely 3-list colorable complete multipartite graphs

Yancai Zhao, Erfang Shan (2010)

Discussiones Mathematicae Graph Theory

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For each vertex v of a graph G, if there exists a list of k colors, L(v), such that there is a unique proper coloring for G from this collection of lists, then G is called a uniquely k-list colorable graph. Ghebleh and Mahmoodian characterized uniquely 3-list colorable complete multipartite graphs except for nine graphs: $K_{2, 2, r}$ r ∈ 4,5,6,7,8, $K_{2, 3, 4}$ , $K_{1 * 4, 4}$ , $K_{1 * 4, 5}$ , $K_{1 * 5, 4}$ . Also, they conjectured that the nine graphs are not U3LC graphs. After that, except for $K_{2, 2, r}$ r ∈ 4,5,6,7,8, the others have been proved not...

Embedding products of graphs into Euclidean spaces

Mikhail Skopenkov (2003)

Fundamenta Mathematicae

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For any collection of graphs $G ₁, . . ., G_{N}$ we find the minimal dimension d such that the product $G ₁ \times . . . \times G_{N}$ is embeddable into $ℝ^{d}$ (see Theorem 1 below). In particular, we prove that (K₅)ⁿ and $(K_{3, 3}) ⁿ$ are not embeddable into $ℝ^{2 n}$ , where K₅ and $K_{3, 3}$ are the Kuratowski graphs. This is a solution of a problem of Menger from 1929. The idea of the proof is a reduction to a problem from so-called Ramsey link theory: we show that any embedding $L k O \to S^{2 n - 1}$ , where O is a vertex of (K₅)ⁿ, has a pair of linked (n-1)-spheres.

On the principal eigencurve of the p-Laplacian related to the Sobolev trace embedding

Abdelouahed El Khalil, Mohammed Ouanan (2005)

Applicationes Mathematicae

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We prove that for any λ ∈ ℝ, there is an increasing sequence of eigenvalues μₙ(λ) for the nonlinear boundary value problem ⎧ $Δ ₚ u = {| u |}^{p - 2} u$ in Ω, ⎨ ⎩ ${| \nabla u |}^{p - 2} \partial u / \partial ν = {λ ϱ (x) | u |}^{p - 2} u + μ {| u |}^{p - 2} u$ on crtial ∂Ω and we show that the first one μ₁(λ) is simple and isolated; we also prove some results about variations of the density ϱ and the continuity with respect to the parameter λ.

About a Pólya-Schiffer inequality

Bodo Dittmar, Maren Hantke (2011)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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For simply connected planar domains with the maximal conformal radius 1 it was proven in 1954 by G. Pólya and M. Schiffer that for the eigenvalues $λ$ of the fixed membrane for any $n$ the following inequality holds $\sum_{k = 1}^{n} \frac{1}{λ_{k}} \geq \sum_{k = 1}^{n} \frac{1}{λ_{k}^{(σ)}},$ where $λ_{k}^{(σ)}$ are the eigenvalues of the unit disk. The aim of the paper is to give a sharper version of this inequality and for the sum of all reciprocals to derive formulas which allow in some cases to calculate exactly this sum.

Edit distance measure for graphs

Tomasz Dzido, Krzysztof Krzywdziński (2015)

Czechoslovak Mathematical Journal

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In this paper, we investigate a measure of similarity of graphs similar to the Ramsey number. We present values and bounds for $g (n, l)$ , the biggest number $k$ guaranteeing that there exist $l$ graphs on $n$ vertices, each two having edit distance at least $k$ . By edit distance of two graphs $G$ , $F$ we mean the number of edges needed to be added to or deleted from graph $G$ to obtain graph $F$ . This new extremal number $g (n, l)$ is closely linked to the edit distance of graphs. Using probabilistic methods we show...

Generalized list colourings of graphs

Mieczysław Borowiecki, Ewa Drgas-Burchardt, Peter Mihók (1995)

Discussiones Mathematicae Graph Theory

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We prove: (1) that $c h_{P} (G) - χ_{P} (G)$ can be arbitrarily large, where $c h_{P} (G)$ and $χ_{P} (G)$ are P-choice and P-chromatic numbers, respectively, (2) the (P,L)-colouring version of Brooks’ and Gallai’s theorems.

Persistency in the Traveling Salesman Problem on Halin graphs

Vladimír Lacko (2000)

Discussiones Mathematicae Graph Theory

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For the Traveling Salesman Problem (TSP) on Halin graphs with three types of cost functions: sum, bottleneck and balanced and with arbitrary real edge costs we compute in polynomial time the persistency partition $E_{A l l}$ , $E_{S o m e}$ , $E_{N o n e}$ of the edge set E, where: $E_{A l l}$ = e ∈ E, e belongs to all optimum solutions, $E_{N o n e}$ = e ∈ E, e does not belong to any optimum solution and $E_{S o m e}$ = e ∈ E, e belongs to some but not to all optimum solutions.

Displaying similar documents to “On graphs whose second least eigenvalue is at least - 1 ”

Displaying similar documents to “On graphs whose second least eigenvalue is at least $- 1$ ”