Hosoya index of unicyclic graphs with prescribed pendent vertices

The Hosoya index z(G) of a (molecular) graph G is defined as the total number of subsets of the edge set, in which any two edges are mutually independent, i.e., the total number of independent-edge sets of G. By G(n, l, k) we denote the set of unicyclic graphs on n vertices with girth and pendent vertices being resp. l and k. Let be the graph obtained by identifying the center of the star S _n-l+1 with any vertex of C _l . By we denote the graph obtained by identifying one pendent vertex of the path P _n-l-k+1 with one pendent vertex of . In this paper, we show that is the unique unicyclic graph with minimal Hosoya index among all graphs in G(n, l, k).      

General properties of some families of graphs defined by systems of equations

In this paper we present a simple method for constructing infinite families of graphs defined by a class of systems of equations over commutative rings. We show that the graphs in all such families possess some general properties including regularity and biregularity, existence of special vertex colorings, and existence of covering maps—hence, embedded spectra—between every two members of the same family. Another general property, recently discovered, is that nearly every graph constructed in this manner edge‐decomposes either the complete, or complete bipartite, graph which it spans. In many instances, specializations of these constructions have proved useful in various graph theory problems, but especially in many extremal problems. A short survey of the related results is included. We also show that the edge‐decomposition property allows one to improve existing lower bounds for some multicolor Ramsey numbers. © 2001 John Wiley & Sons, Inc. J Graph Theory 38: 65–86, 2001     

Binary Optimal Odd Formally Self-Dual Codes

In this paper, we study binary optimal odd formallyself-dual codes. All optimal odd formally self-dual codes areclassified for length up to 16. The highest minimum weight ofany odd formally self-dual codes of length up to 24 is determined. We also show that there is a unique linearcode for parameters [16, 8, 5] and [22, 11, 7], up to equivalence.     

Gallai's Conjecture For Graphs of Girth at Least Four

In 1966, Gallai conjectured that for any simple, connected graph G having n vertices, there is a path‐decomposition of G having at most paths. In this article, we show that for any simple graph G having girth , there is a path‐decomposition of G having at most paths, where is the number of vertices of odd degree in G and is the number of nonisolated vertices of even degree in G.     

Parity and disparity subgraphs

A parity subgraph of a graph is a spanning subgraph such that the degrees of each vertex have the same parity in both the subgraph and the original graph. Known results include that every graph has an odd number of minimal parity subgraphs. Define a disparity subgraph to be a spanning subgraph such that each vertex has degrees of opposite parities in the subgraph and the original graph. (Only graphs with all even-order components can have disparity subgraphs). Every even-order spanning tree contains both a unique parity subgraph and a unique disparity subgraph. Moreover, every minimal disparity subgraph is shown to be paired by sharing a spanning tree with an odd number of minimal parity subgraphs, and every minimal parity subgraph is similarly paired with either one or an even number of minimal disparity subgraphs.     

New upper bounds on linear coloring of planar graphs

A proper vertex coloring of a graph G is linear if the graph induced by the vertices of any two color classes is the union of vertex-disjoint paths. The linear chromatic number lc(G) of the graph G is the smallest number of colors in a linear coloring of G. In this paper, it is proved that every planar graph G with girth g and maximum degree Δ has(1)lc(G) ≤Δ 21 if Δ≥ 9; (2)lc(G) ≤「Δ/2」 + 7 ifg ≥ 5; (3) lc(G) ≤「Δ/2」 + 2 ifg ≥ 7 and Δ≥ 7.     

不含短圈的平面图的2- 距离染色

图的2- 距离染色是将图中距离不超过2 的点对染不同的色. 本文证明了g(G) > 5 且Δ(G) > 18的平面图G 有(Δ + 6)-2- 距离染色.     

Measurement technique for characterizing odd aberration of lithographic projection optics based on dipole illumination

A dipole illumination based measurement technique for measuring odd aberration of lithographic projection optics is proposed. In the present technique, odd aberration is extracted from the image displacements at multiple illumination settings. Theoretical analysis of the impact of illumination profile on the measurement accuracy is presented. By use of dipole illumination, the image displacement variation range that determines the measurement accuracy is enlarged. Using lithographic simulator PROLITH, the measurement accuracies of odd aberration under conventional illumination and dipole illumination are compared. The simulation results show that the measurement accuracy of odd aberration increases significantly when dipole illumination is used.