Цветовая энергия некоторых кластерных графов

Let $G$ be a simple connected graph. The energy of a graph $G$ is defined as sum of the absolute eigenvalues of an adjacency matrix of the graph $G$. It represents a proper generalization of a formula valid for the total $\pi$-electron energy of a conjugated hydrocarbon as calculated by the Huckel molecular orbital (HMO) method in quantum chemistry. A coloring of a graph $G$ is a coloring of its vertices such that no two adjacent vertices share the same color. The minimum number of colors needed for the coloring of a graph $G$ is called the chromatic number of $G$ and is denoted by $\chi(G)$. The color energy of a graph $G$ is defined as the sum of absolute values of the color eigenvalues of $G$. The graphs with large number of edges are referred as cluster graphs. Cluster graphs are graphs obtained from complete graphs by deleting few edges according to some criteria. It can be obtained on deleting some edges incident on a vertex, deletion of independent edges/triangles/cliques/path P3 etc. Bipartite cluster graphs are obtained by deleting few edges from complete bipartite graphs according to some rule. In this paper, the color energy of cluster graphs and bipartite cluster graphs are studied.

On Total Vertex Irregularity Strength of Hexagonal Cluster Graphs

For a simple graph G with a vertex set V G and an edge set E G , a labeling f : V G ∪ ​ E G ⟶ 1,2 , ⋯ , k is called a vertex irregular total k − labeling of G if for any two different vertices x and y in V G we have w t x ≠ w t y where w t x = f x + ∑ u ∈ V G f x u . The smallest positive integer k such that G has a vertex irregular total k − labeling is called the total vertex irregularity strength of G , denoted by tvs G . The lower bound of tvs G for any graph G have been found by Baca et. al. In this paper, we determined the exact value of the total vertex irregularity strength of the hexagonal cluster graph on n cluster for n ≥ 2 . Moreover, we show that the total vertex irregularity strength of the hexagonal cluster graph on n cluster is 3 n 2 + 1 / 2 .

Features-Based Deisotoping Method for Tandem Mass Spectra

For high-resolution tandem mass spectra, the determination of monoisotopic masses of fragment ions plays a key role in the subsequent peptide and protein identification. In this paper, we present a new algorithm for deisotoping the bottom-up spectra. Isotopic-cluster graphs are constructed to describe the relationship between all possible isotopic clusters. Based on the relationship in isotopic-cluster graphs, each possible isotopic cluster is assessed with a score function, which is built by combining nonintensity and intensity features of fragment ions. The non-intensity features are used to prevent fragment ions with low intensity from being removed. Dynamic programming is adopted to find the highest score path with the most reliable isotopic clusters. The experimental results have shown that the average Mascot scores and F-scores of identified peptides from spectra processed by our deisotoping method are greater than those by YADA and MS-Deconv software.