Eccentricity energy change of complete multipartite graphs due to edge deletion

The eccentricity matrix ɛ(G) of a graph G is obtained from the distance matrix of G by retaining the largest distances in each row and each column, and leaving zeros in the remaining ones. The eccentricity energy of G is sum of the absolute values of the eigenvalues of ɛ(G). Although the eccentricity matrices of graphs are closely related to the distance matrices of graphs, a number of properties of eccentricity matrices are substantially different from those of the distance matrices. The change in eccentricity energy of a graph due to an edge deletion is one such property. In this article, we give examples of graphs for which the eccentricity energy increase (resp., decrease) but the distance energy decrease (resp., increase) due to an edge deletion. Also, we prove that the eccentricity energy of the complete k-partite graph Kn 1, ... , nk with k ≥ 2 and ni ≥ 2, increases due to an edge deletion.

Biconed Graphs, Weighted Forests, and $h$-Vectors of Matroid Complexes

A well-known conjecture of Richard Stanley posits that the $h$-vector of the independence complex of a matroid is a pure ${\mathcal O}$-sequence. The conjecture has been established for various classes but is open for graphic matroids. A biconed graph is a graph with two specified 'coning vertices', such that every vertex of the graph is connected to at least one coning vertex. The class of biconed graphs includes coned graphs, Ferrers graphs, and complete multipartite graphs. We study the $h$-vectors of graphic matroids arising from biconed graphs, providing a combinatorial interpretation of their entries in terms of '$2$-weighted forests' of the underlying graph. This generalizes constructions of Kook and Lee who studied the Möbius coinvariant (the last nonzero entry of the $h$-vector) of graphic matroids of complete bipartite graphs. We show that allowing for partially $2$-weighted forests gives rise to a pure multicomplex whose face count recovers the $h$-vector, establishing Stanley's conjecture for this class of matroids. We also discuss how our constructions relate to a combinatorial strengthening of Stanley's Conjecture (due to Klee and Samper) for this class of matroids.

On the local irregularity vertex coloring of volcano, broom, parachute, double broom and complete multipartite graphs

Let [Formula: see text] be a simple, finite, undirected, and connected graph with vertex set [Formula: see text] and edge set [Formula: see text]. A bijection [Formula: see text] is label function [Formula: see text] if [Formula: see text] and for any two adjacent vertices [Formula: see text] and [Formula: see text], [Formula: see text] where [Formula: see text] and [Formula: see text] is set ofvertices adjacent to [Formula: see text]. [Formula: see text] is called local irregularity vertex coloring. The minimum cardinality of local irregularity vertex coloring of [Formula: see text] is called chromatic number local irregular denoted by [Formula: see text]. In this paper, we verify the exact values of volcano, broom, parachute, double broom and complete multipartite graphs.

Nearly Gorenstein Rings Arising from Finite Graphs

The classification of complete multipartite graphs whose edge rings are nearly Gorenstein as well as that of finite perfect graphs whose stable set rings are nearly Gorenstein is achieved.

Indirect identification of horizontal gene transfer

Several implicit methods to infer horizontal gene transfer (HGT) focus on pairs of genes that have diverged only after the divergence of the two species in which the genes reside. This situation defines the edge set of a graph, the later-divergence-time (LDT) graph, whose vertices correspond to genes colored by their species. We investigate these graphs in the setting of relaxed scenarios, i.e., evolutionary scenarios that encompass all commonly used variants of duplication-transfer-loss scenarios in the literature. We characterize LDT graphs as a subclass of properly vertex-colored cographs, and provide a polynomial-time recognition algorithm as well as an algorithm to construct a relaxed scenario that explains a given LDT. An edge in an LDT graph implies that the two corresponding genes are separated by at least one HGT event. The converse is not true, however. We show that the complete xenology relation is described by an rs-Fitch graph, i.e., a complete multipartite graph satisfying constraints on the vertex coloring. This class of vertex-colored graphs is also recognizable in polynomial time. We finally address the question “how much information about all HGT events is contained in LDT graphs” with the help of simulations of evolutionary scenarios with a wide range of duplication, loss, and HGT events. In particular, we show that a simple greedy graph editing scheme can be used to efficiently detect HGT events that are implicitly contained in LDT graphs.