Edge-Complement Graphs – Another Approach (original) (raw)

On the Generalized Complement of Some Graphs

Asia Pacific Journal of Mathematics, 2021

A. In this paper we study the generalized complement of the graph G m,n = (V, E) for some values of m, n. We study the generalized complement of G m,n graphs with respect to the equal degree partition. The 2−complement of G m,n graphs are also determined for m = 2, n is even or odd. In particular, for some values of m, n ∈ N, we studied the complement of G m,n graphs with respect to the equal degree partition and the 2−complement of G m,n graphs. We determine the partitions P k , k ∈ N of the vertex set V such that the generalized complement of G m,n graph is a path graph and a comb graph.

Edge connectivity in difference graphs and some new constructions of partial sum families

European Journal of Combinatorics, 2011

In this paper, bounds for the edge connectivity of m-Cayley graphs are found, and also several structural conditions are given for a connected k-regular bi-abelian graph to have edge connectivity strictly less than k. Finally, two infinite families of partial sum families that generate directed strongly regular graphs with new parameters are shown.

Characterization of Generalized Complements of a Graph

Advances in Mathematics: Scientific Journal, 2020

} be a partition of vertex set V (G) of order k ≥ 2. For all V i and V j in P , i = j, remove the edges between V i and V j in graph G and add the edges between V i and V j which are not in G. The graph G P k thus obtained is called the k−complement of graph G with respect to the partition P. For each set V r in P , remove the edges of graph G inside V r and add the edges of G (the complement of G) joining the vertices of V r. The graph G P k(i) thus obtained is called the k(i)−complement of graph G with respect to the partition P. In this paper, we characterize few properties of generalized complements of a graph.

Edge-antimagic graphs

Discrete Mathematics, 2007

, form an arithmetic progression starting from a and having common difference d. An (a, d)-edge-antimagic total labeling is called super (a, d)-edge-antimagic total if g(V (G)) = {1, 2, . . . , |V (G)|}. We study super (a, d)-edge-antimagic properties of certain classes of graphs, including friendship graphs, wheels, fans, complete graphs and complete bipartite graphs.

The edge-flipping group of a graph

2009

Let X=(V,E) be a finite simple connected graph with n vertices and m edges. A configuration is an assignment of one of two colors, black or white, to each edge of X. A move applied to a configuration is to select a black edge ϵ∈ E and change the colors of all adjacent edges of ϵ. Given an initial configuration and a final configuration, try to find a sequence of moves that transforms the initial configuration into the final configuration. This is the edge-flipping puzzle on X, and it corresponds to a group action. This group is called the edge-flipping group W_E(X) of X. This paper shows that if X has at least three vertices, W_E(X) is isomorphic to a semidirect product of (Z/2Z)^k and the symmetric group S_n of degree n, where k=(n-1)(m-n+1) if n is odd, k=(n-2)(m-n+1) if n is even, and Z is the additive group of integers.

Certain classes of complementary equienergetic graphs

Transactions on Combinatorics, 2021

The energy of a graph is the sum of the absolute values of the eigenvalues of a graph. Two graphs are said to be equienergetic if they have same energy. A graph is said to be complementary equienergetic if it is equienergetic with its complement. Recently several complementary equienergetic graphs have been identified. In this paper, we characterize the cycle, path, complete bipartite regular graph and iterated line graph of regular graph, which are complementary equienergetic.

A graph and its complement with specified properties I: connectivity

International Journal of Mathematics and Mathematical Sciences, 1979

Dedicated to Karl Menger ABSTRACT. We investigate the conditions under which both a graph G and its complement G possess a specified property. In particular, we characterize all graphs G for which G and G both (a) have connectivity one, (b) have line-connectivity one, (c) are 2-connected, (d) are forests, (e) are bipartite, (f) are outerplanar and (g) are eulerlan. The proofs are elementary but amusing. KEF WORDS AND PHRASES. Graphs, Complement. AMS (MOS) SUBJECT CLASSIFICATION (1970) CODES. 05C99.

On the Complementary Equienergetic Graphs

arXiv: Combinatorics, 2019

Energy of a simple graph GGG, denoted by mathcalE(G)\mathcal{E}(G)mathcalE(G), is the sum of the absolute values of the eigenvalues of GGG. Two graphs with the same order and energy are called equienergetic graphs. A graph GGG with the property GcongoverlineGG\cong \overline{G}GcongoverlineG is called self-complementary graph, where overlineG\overline{G}overlineG denotes the complement of GGG. Two non-self-complementary equienergetic graphs G1G_1G1 and G2G_2G2 satisfying the property G1congoverlineG2G_1\cong \overline{G_2}G_1congoverlineG_2 are called complementary equienergetic graphs. Recently, Ramane et al. [Graphs equienergetic with their complements, MATCH Commun. Math. Comput. Chem. 82 (2019) 471-480] initiated the study of the complementary equienergetic regular graphs and they asked to study the complementary equienergetic non-regular graphs. In this paper, by developing some computer codes and by making use of some software like Nauty, Maple and GraphTea, all the complementary equienergetic graphs with at most 10 vertices as well as all the members of the graph class $\...

Characterizations of some parity signed graphs

Australas. J Comb., 2021

We describe parity labelings of signed graphs; equivalently, cuts of the underlying graph that have nearly equal sides. We characterize the balanced signed graphs which are parity signed graphs. We give structural characterizations of all parity signed stars, bistars, cycles, paths and complete bipartite graphs. The rna number of a graph is the smallest cut size that has nearly equal sides; we find it for a few classes of parity signed graphs.