Theorem 2 of [2], which claims to settle Atteson’s edge radius conjecture [1], is invalidly proven. The argument in [2] is inductive, and is based on the assumption that if initially ||D̂ − DT ||∞ < l(e) 4 then as the algorithm proceeds, the intermediate distance matrices obtained by agglomeration are within l∞ distance l(e) 4 from some tree metric (this point is in fact not explicitly stated in...

In this paper, we study the behaviour of the generalized power domination number of a graph by small changes on the graph, namely edge and vertex deletion and edge contraction. We prove optimal bounds for γP,k(G − e), γP,k(G/e) and for γP,k(G − v) in terms of γP,k(G), and give examples for which these bounds are tight. We characterize all graphs for which γP,k(G− e) = γP,k(G) + 1 for any edge e...

A graph is minimally $k$-connected ($k$-edge-connected) if it and deleting any arbitrary chosen edge always leaves a which not ($k$-edge-connected). Let $m= \binom{d}{2}+t$, $1\leq t\leq d$ $G_m$ be the obtained from complete $K_d$ by adding one new vertex of degree $t$. $H_m$ $K_d\backslash\{e\}$ adjacent to precisely two vertices $d-1$ in $K_d\backslash\{e\}$. Rowlinson [Linear Algebra Appl.,...

For understanding of the physics processes determining the radial profiles of the plasma pressure in the advanced tokamak plasmas, correlation between the edge and the internal transport barriers (ETB and ITB) has been studied. We fond that the edge pedestal beta, βp-ped, increases almost linearly with the total βp over a wide range of the plasma current for the type I ELMing H-mode, and the de...

Let G be a mixed graph and L(G) be the Laplacian matrix of G. In this paper, the coefficients of the Laplacian characteristic polynomial of G are studied. The first derivative of the characteristic polynomial of L(G) is explicitly expressed by means of Laplacian characteristic polynomials of its edge deleted subgraphs. As a consequence, it is shown that the Laplacian characteristic polynomial o...

Context. Edge-on spiral galaxies often have stellar disks with relatively sharp truncations and there is extensive observational material of warped HI-layers in the outer parts. Warps appear to start preferentially near the optical boundaries of the disks, but the possible relation between truncations and warps has not been studied in detail. Aims. The aim is to make a comparative study of warp...

A Graph is called 2-self-centered if its diameter and radius both equal to 2. In this paper, we begin characterizing these graphs by characterizing edge-maximal 2-self-centered graphs via their compliments. Then we split characterizing edge-minimal 2-self-centered graphs into two cases. First, we characterize edge-minimal 2-self-centered graphs without triangles by introducing intersecting bi-i...

In 1962, Erdős gave a sufficient condition for Hamilton cycles in terms of the vertex number, edge number, and minimum degree of graphs which generalized Ore’s theorem. One year later, Moon and Moser gave an analogous result for Hamilton cycles in balanced bipartite graphs. In this paper we present the spectral analogues of Erdős’ theorem and Moon-Moser’s theorem, respectively. Let Gk n be the ...