Spectral Bounds for the Connectivity of Regular Graphs with Given Order

The second-largest eigenvalue and second-smallest Laplacian eigenvalue of a graph are measures of its connectivity. These eigenvalues can be used to analyze the robustness, resilience, and synchronizability of networks, and are related to connectivity attributes such as the vertex- and edge-connectivity, isoperimetric number, and characteristic path length. In this paper, we present two upper bounds for the second-largest eigenvalues of regular graphs and multigraphs of a given order which guarantee a desired vertex- or edge-connectivity. The given bounds are in terms of the order and degree of the graphs, and hold with equality for infinite families of graphs. These results answer a question of Mohar.Comment: 24 page

Criteria for SLOCC and LU Equivalence of Generic Multi-qudit States

In this paper, we study the stochastic local operation and classical communication (SLOCC) and local unitary (LU) equivalence for multi-qudit states by mode-$n$ matricization of the coefficient tensors. We establish a new scheme of using the CANDECOMP/PARAFAC (CP) decomposition of tensors to find necessary and sufficient conditions between the mode-$n$ unfolding and SLOCC\&LU equivalence for pure multi-qudit states. For multipartite mixed states, we present a necessary and sufficient condition for LU equivalence and necessary condition for SLOCC equivalence

Two Ramsey-related Problems

Extremal combinatorics is one of the central branches of discrete mathematics and has experienced an impressive growth during the last few decades. It deals with the problem of determining or estimating the maximum or minimum possible size of a combinatorial structure which satisfies certain requirements. In this dissertation, we focus on studying the minimum number of edges of certain co-critical graphs. Given an integer r ≥ 1 and graphs G; H1; : : : ;Hr, we write → G (H1; : : : ;Hr) if every r-coloring of the edges of G contains a monochromatic copy of Hi in color i for some i ϵ {1; : : : ; r}. A non-complete graph G is (H1; : : : ;Hr)-co-critical if -/ \u3e (H1; : : : ;Hr), but G + uv → (H1; : : : ;Hr) for every pair of non-adjacent vertices u; v in G. Motivated in part by Hanson and Toft\u27s conjecture from 1987, we study the minimum number of edges over all (Kt; Tk)-co-critical graphs on n vertices, where Tk denotes the family of all trees on k vertices. We apply graph bootstrap percolation on a not necessarily Kt-saturated graph to prove that for all t ≥ 4 and k ≥ max{6, t}, there exists a constant c(t, k) such that, for all n ≥ (t - 1)(k - 1) + 1, if G is a (Kt; Tk)-co-critical graph on n vertices, then e(G) ≥ (4t-9/2 + 1/2 [K/2]) n - c _t, k). We then show that this is asymptotically best possible for all sufficiently large n when t ϵ {4, 5} and k ≥ 6. The method we developed may shed some light on solving Hanson and Toft\u27s conjecture, which is wide open. We also study Ramsey numbers of even cycles and paths under Gallai colorings, where a Gallai coloring is a coloring of the edges of a complete graph without rainbow triangles, and a Gallai k-coloring is a Gallai coloring that uses at most k colors. Given an integer k ≥ 1 and graphs H1, : : : ,Hk, the Gallai-Ramsey number GR(H1; : : : ;Hk) is the least integer n such that every Gallai k-coloring of the complete graph Kn contains a monochromatic copy of Hi in color i for some i ϵ {1; : : : ; k}. We completely determine the exact values of GR(H1; : : : ;Hk) for all k ≥ 2 when each Hi is a path or an even cycle on at most 13 vertices

Poly[μ2-hydroxido-μ4-sulfato-neodym­ium(III)]

The title compound, [Nd(OH)(SO4)]n, was obtained hydro­thermally from an aqueous solution of neodymium nitrate, 1,2-propane­diamine and sulfuric acid. The structure features nona­coordinated neodymium with sulfate and hydroxide anions acting as bridging ligands. The OH group forms a weak O—H⋯O hydrogen bond with an O⋯O distance of 3.224 (5) Å