SOME PROPERTIES OF GLUED GRAPHS AT COMPLETE CLONE IN THE VIEW OF ALGEBRAIC COMBINATORICS.
A glued graph at complete clone is obtained from combining two graphs by identifying edges of of each original graph. We investigate how to change some properties such as height big height Krull dimension Betti numbers by gluing of two graphs at complete clone. We give a sufficient and necessary condition so that the glued graph of tw o Cohen-Macaulay chordal graphs at complete clone is a Cohen-Macaulay graph. Moreover we present the conditions that the edge ideal of gluing of two graphs at complete clone has linear resolution whenever the edge ideals of original graphs have linear resolution. We show when gluing of two independence complexes line graphs complement graphs can be expressed as independence complex line graph and complement of gluing of two graphs.
Key Words: Glued graph Height Big height Krull dimension Projective dimension Linear resolution Betti number Cohen-Macaulay.
The concept edge ideal was first introduced by Villarreal in  that is let be a simple (no loops or multiple edges) graph on the vertex set and the edge set . Associate to is a quadratic square free monomial ideal with a field which is generated by such that . An approach to studying combinatorial properties of a graph is to examine some of algebraic invariants of the edge ideal. Indeed an aim of recent much research has been to create a dictionary between algebraic properties of and properties of .
In 2003 Uiyyasathain presented a new class of graphs in  glued graphs that is let and be any graphs and be non-trivial connected and such that with an isomorphism . The glued graph of and at and with respect to denoted by is as the graph that results from combining with by identifying and in the glued graph. In  Promsakon and Uiyyasathain characterized graph gluing between trees forests and bipartite graphs. Also they could give an upper bound of the chromatic number of glued graphs in terms of their original graphs. In  Uiyyasathain and Saduakdee studied the perfection of glued graphs at -clone. In  Uiyyasathain and Jongthawonwuth obtained bounds of the clique partition numbers of glued graph at -clones and -clones in terms of their original graphs. In  Pimpasalee and Uiyyasathain investigated bounds of clique covering numbers of glued graphs at -clones in terms of their original graphs.
As mentioned above the study of glued graphs from combinatorial points of view has become an active area but our main purpose of the current paper is to express algebraic features of glued graphs at complete clone using combinatorial properties. Also we have tried as much as possible to give an accurate description of some properties of a glued graph in terms of their original graphs. Furthermore we intend to verify whether the property of being Cohen- Macaulay Gorenstein (for chordal graphs) having linear resolution transfer from the glued graph to original graphs and vice versa. One of the main reasons for the importance of gluing of two graphs is the fact that this operation creates a larger class of graphs which one can obtain the results on the larger graph according to the information of the smaller graphs.
Our paper is organized as follows. In section 2 we give an explicit formula for computing the height of glued graph at complete clone. Also we present a lower bound for the big height of the glued graph at complete clone and characterize the glued graphs satisfying such bound. In section 3 we provide a necessary and sufficient condition which the equality holds for any We obtain explicit formulas for computing and also we present a lower bound for . We show that having linear resolution of the edge ideal of glued graph at complete clone implies that edge ideals of original graphs have linear resolution. A simple example illustrates having linear resolution the edge ideals of original graphs does not guarantee the existence of this property of the edge ideal of glued graph at complete clone then we present a necessary and sufficient condition for having linear resolution of the edge ideal of glued graph at complete clone.
For chordal graphs we provide the conditions that being Cohen- Macaulay preserves under operation gluing of two graphs at complete clone and vice versa. As a result we obtain a necessary and sufficient condition for being Gorenstein of gluing of two Cohen-Macaulay chordal graphs at complete clone. In Section 4 we give an upper bound for the projective dimension and Alexander dual of the edge ideal of the glued graph at any clone. For any two connected graphs containing a connected subgraph H we investigate the relation between the complement line graph and independence complex of the glued graph at clone and the complement line graph and independence complex of original graphs. Furthermore we determine a sufficient condition for vertex decomposability of the glued graph at any clone. In  it is proved that the glued graph of connected chordal graphs is chordal. The converse is not true in general.
We characterize a useful condition for being chordal of original graphs when the glued graph is chordal.
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