The number of removable edges in a 4-connected graph

AbstractLet G be a 4-connected graph. For an edge e of G, we do the following operations on G: first, delete the edge e from G, resulting the graph G−e; second, for all the vertices x of degree 3 in G−e, delete x from G−e and then completely connect the 3 neighbors of x by a triangle. If multiple edges occur, we use single edges to replace them. The final resultant graph is denoted by G⊖e. If G⊖e is still 4-connected, then e is called a removable edge of G. In this paper we prove that every 4-connected graph of order at least six (excluding the 2-cyclic graph of order six) has at least (4|G|+16)/7 removable edges. We also give the structural characterization of 4-connected graphs for which the lower bound is sharp

Removable edges in a k-connected graph and a construction method for k-connected graphs

AbstractAn edge e of a k-connected graph G is said to be a removable edge if G⊖e is still k-connected, where G⊖e denotes the graph obtained from G by deleting e to get G−e and, for any end vertex of e with degree k−1 in G−e, say x, deleting x and then adding edges between any pair of non-adjacent vertices in NG−e(x). Xu and Guo [Liqiong Xu, Xiaofeng Guo, Removable edges in a 5-connected graph and a construction method of 5-connected graphs, Discrete Math. 308 (2008) 1726–1731] proved that a 5-connected graph G has no removable edge if and only if G≅K6, using this result, they gave a construction method for 5-connected graphs. A k-connected graph G is said to be a quasi (k+1)-connected if G has no nontrivial k-vertex cut. Jiang and Su [Hongxing Jiang, Jianji Su, Minimum degree of minimally quasi (k+1)-connected graphs, J. Math. Study 35 (2002) 187–193] conjectured that for k≥4 the minimum degree of a minimally quasi k-connected graph is equal to k−1. In the present paper, we prove this conjecture and prove for k≥3 that a k-connected graph G has no removable edge if and only if G is isomorphic to either Kk+1 or (when k is even) the graph obtained from Kk+2 by removing a 1-factor. Based on this result, a construction method for k-connected graphs is given

J. Phys. Chem. B

The effect of pressure on the structure, interionic interactions, and properties of the ionic liquid (IL) 1-butyl-3-methylimidazolium hexafluorophosphate ([C(4)mim]-[PF6]) was studied. using an all-atom molecular dynamics simulation. A distinct conformational transition from anti (a) to gauche (g) form based on the deformation of the first C-C bond of the butyl chain was observed under high pressure, and the ratio of the a conformation that changed into the g conformation was 5.5% at 6000 bar. Under high pressure, the configuration of the a and g conformer for [C(4)mim](+) tends to make the alkyl chain distorted to the inside of the ring. Results on the density changes indicate a small increase from 5000 to 6000 bar, which could be attributed to the writhing of the reducing end of the alkyl chain in the cation at higher pressure. These simulation results are well agreed with the experimental results. Transport properties were also calculated at different pressures. The results show that diffusion of the ions is reduced under high pressure, and the viscosity is dramatically enhanced.The effect of pressure on the structure, interionic interactions, and properties of the ionic liquid (IL) 1-butyl-3-methylimidazolium hexafluorophosphate ([C(4)mim]-[PF6]) was studied. using an all-atom molecular dynamics simulation. A distinct conformational transition from anti (a) to gauche (g) form based on the deformation of the first C-C bond of the butyl chain was observed under high pressure, and the ratio of the a conformation that changed into the g conformation was 5.5% at 6000 bar. Under high pressure, the configuration of the a and g conformer for [C(4)mim](+) tends to make the alkyl chain distorted to the inside of the ring. Results on the density changes indicate a small increase from 5000 to 6000 bar, which could be attributed to the writhing of the reducing end of the alkyl chain in the cation at higher pressure. These simulation results are well agreed with the experimental results. Transport properties were also calculated at different pressures. The results show that diffusion of the ions is reduced under high pressure, and the viscosity is dramatically enhanced