The edge C₄ graph of a graph G, E₄(G) is a graph whose vertices are the edges of G and two vertices in E₄(G) are adjacent if the corresponding edges in G are either incident or are opposite edges of some C₄. In this paper, we show that there exist infinitely many pairs of non isomorphic graphs whose edge C₄ graphs are isomorphic. We study the relationship between the diameter, radius and domination number of G and those of E₄(G). It is shown that for any graph G without isolated vertices, there exists a super graph H such that C(H) = G and C(E₄(H)) = E₄(G). Also we give forbidden subgraph characterizations for E₄(G) being a threshold graph, block graph, geodetic graph and weakly geodetic graph.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1499,
author = {Manju K. Menon and A. Vijayakumar},
title = {The edge C4 graph of some graph classes},
journal = {Discussiones Mathematicae Graph Theory},
volume = {30},
year = {2010},
pages = {365-375},
zbl = {1217.05194},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1499}
}
Manju K. Menon; A. Vijayakumar. The edge C₄ graph of some graph classes. Discussiones Mathematicae Graph Theory, Tome 30 (2010) pp. 365-375. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1499/
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