On Omega Index and Average Degree of Graphs

Average degree of a graph is defined to be a graph invariant equal to the arithmetic mean of all vertex degrees and has many applications, especially in determining the irregularity degrees of networks and social sciences. In this study, some properties of average degree have been studied. Effect of vertex deletion on this degree has been determined and a new proof of the handshaking lemma has been given. Using a recently defined graph index called o m e g a index, average degree of trees, unicyclic, bicyclic, and tricyclic graphs have been given, and these have been generalized to k -cyclic graphs. Also, the effect of edge deletion has been calculated. The average degree of some derived graphs and some graph operations have been determined.

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.

A Parameterized Complexity View on Collapsing k-Cores

We study the -hard graph problem Collapsed k-Core where, given an undirected graph G and integers b, x, and k, we are asked to remove b vertices such that the k-core of remaining graph, that is, the (uniquely determined) largest induced subgraph with minimum degree k, has size at most x. Collapsed k-Core was introduced by Zhang et al. (2017) and it is motivated by the study of engagement behavior of users in a social network and measuring the resilience of a network against user drop outs. Collapsed k-Core is a generalization of r-Degenerate Vertex Deletion (which is known to be -hard for all r ≥ 0) where, given an undirected graph G and integers b and r, we are asked to remove b vertices such that the remaining graph is r-degenerate, that is, every its subgraph has minimum degree at most r. We investigate the parameterized complexity of Collapsed k-Core with respect to the parameters b, x, and k, and several structural parameters of the input graph. We reveal a dichotomy in the computational complexity of Collapsed k-Core for k ≤ 2 and k ≥ 3. For the latter case it is known that for all x ≥ 0 Collapsed k-Core is -hard when parameterized by b. For k ≤ 2 we show that Collapsed k-Core is -hard when parameterized by b and in when parameterized by (b + x). Furthermore, we outline that Collapsed k-Core is in when parameterized by the treewidth of the input graph and presumably does not admit a polynomial kernel when parameterized by the vertex cover number of the input graph.