On weighted spectral radius of unraveled balls and normalized Laplacian eigenvalues

For a graph G, the unraveled ball of radius r centered at a vertex v is the ball of radius r centered at v in the universal cover of G. We obtain a lower bound on the weighted spectral radius of unraveled balls of fixed radius in a graph with positive weights on edges, which is used to present an upper bound on the $s^{\text{th}}$ (where $s\ge 2$) smallest normalized Laplacian eigenvalue of irregular graphs under minor assumptions. Moreover, when $s=2$, the result may be regarded as an Alon–Boppana type bound for a class of irregular graphs.     

Connected cubic s-arc-regular Cayley graphs of finite nonabelian simple groups

A graph is said to be s-arc-regular if its full automorphism group acts regularly on the set of its s-arcs. In this paper, we investigate connected cubic s-arc-regular Cayley graphs of finite nonabelian simple groups. Two suffcient and necessary conditions for such graphs to be 1- or 2-arc-regular are given and based on the conditions, several infinite families of 1-or 2-arc-regular cubic Cayley graphs of alternating groups are constructed.     

The number of spanning trees in an (r,s)‐semiregular graph and its line graph

For a graph G, a “spanning tree” in G is a tree that has the same vertex set as G. The number of spanning trees in a graph (network) G, denoted by t(G), is an important invariant of the graph (network) with lots of decisive applications in many disciplines. In the article by Sato (Discrete Math. 2007, 307, 237), the number of spanning trees in an (r, s)‐semiregular graph and its line graph are obtained. In this article, we give short proofs for the formulas without using zeta functions. Furthermore, by applying the formula that enumerates the number of spanning trees in the line graph of an (r, s)‐semiregular graph, we give a new proof of Cayley's Theorem. © 2013 Wiley Periodicals, Inc.     

The reduced expressions in a Coxeter system with a strictly complete Coxeter graph

Let (W,S)$(W,S)$ be a Coxeter system with a strictly complete Coxeter graph. The present paper concerns the set Red(z)$\mathrm{Red}(z)$ of all reduced expressions for any z∈W$z\in W$. By associating each bc-expression to a certain symbol, we describe the set Red(z)$\mathrm{Red}(z)$ and compute its cardinal |Red(z)|$|\mathrm{Red}(z)|$ in terms of symbols. An explicit formula for |Red(z)|$|\mathrm{Red}(z)|$ is deduced, where the Fibonacci numbers play a crucial role.     

Matching energy of unicyclic and bicyclic graphs with a given diameter

Gutman and Wagner proposed the concept of matching energy (ME) and pointed out that the chemical applications of ME go back to the 1970s. Let G be a simple graph of order n and be the roots of its matching polynomial. The ME of G is defined to be the sum of the absolute values of . In this article, we characterize the graphs with minimal ME among all unicyclic and bicyclic graphs with a given diameter d. © 2014 Wiley Periodicals, Inc. Complexity 21: 224–238, 2015     

Optimal Groomings with Grooming Ratios Six and Seven

Grooming uniform all‐to‐all traffic in optical (SONET) rings with grooming ratio C requires the determination of a decomposition of the complete graph into subgraphs each having at most C edges. The drop cost of such a grooming is the total number of vertices of nonzero degree in these subgraphs, and the grooming is optimal when the drop cost is minimum. The determination of optimal C‐groomings has been considered for , and completely solved for . For , it has been shown that the lower bound for the drop cost of an optimal C‐grooming can be attained for almost all orders with 5 exceptions and 308 possible exceptions. For , there are infinitely many unsettled orders; especially the case is far from complete. In this paper, we show that the lower bound for the drop cost of a 6‐grooming can be attained for almost all orders by reducing the 308 possible exceptions to 3, and that the lower bound for the drop cost of a 7‐grooming can be attained for almost all orders with seven exceptions and 16 possible exceptions. Moreover, for the unsettled orders, we give upper bounds for the minimum drop costs.     

Formation of lithium,sodium and potassium complexes with 1,3,5‐triamino‐1,3,5‐trideoxy‐cis‐inositol (taci)

Single crystals of (1,3‐diamino‐5‐azaniumyl‐1,3,5‐trideoxy‐cis‐inositol‐κ³O²,O⁴,O⁶)(1,3,5‐triamino‐1,3,5‐trideoxy‐cis‐inositol‐κ³O²,O⁴,O⁶)lithium(I) diiodide dihydrate, [Li(C₆H₁₆N₃O₃)(C₆H₁₅N₃O₃)]I₂·2H₂O or [Li(Htaci)(taci)]I₂·2H₂O (taci is 1,3,5‐triamino‐1,3,5‐trideoxy‐cis‐inositol), (I), bis(1,3,5‐triamino‐1,3,5‐trideoxy‐cis‐inositol‐κ³O²,O⁴,O⁶)sodium(I) iodide, [Na(C₆H₁₅N₃O₃)₂]I or [Na(taci)₂]I, (II), and bis(1,3,5‐triamino‐1,3,5‐trideoxy‐cis‐inositol‐κ³O²,O⁴,O⁶)potassium(I) iodide, [K(C₆H₁₅N₃O₃)₂]I or [K(taci)₂]I, (III), were grown by diffusion of MeOH into aqueous solutions of the complexes. The structures of the Na and K complexes are isotypic. In all three complexes, the taci ligands adopt a chair conformation with axial hydroxy groups, and the metal cations exhibit exclusive O‐atom coordination. The six O atoms of the resulting MO₆ unit define a centrosymmetric trigonal antiprism with approximate D_3d symmetry. The interligand O...O distances increase significantly in the order Li < Na < K. The structure of (I) exhibits a complex three‐dimensional network of R—NH₂—H...NH₂—R, R—O—H...NH₂—R and R—O—H...O(H)—H...NH₂—R hydrogen bonds. The structures of the Na and K complexes consist of a stack of layers, in which each taci ligand is bonded to three neighbours via pairwise O—H...NH₂ interactions between vicinal HO—CH—CH—NH₂ groups.