Calculating the Number of vertices Labeled Order Six Disconnected Graphs which Contain Maximum Seven Loops and Even Number of Non-loop Edges Without Parallel Edges

AbstractWe investigate thegroup irregularity strength(sg(G)) of graphs, i.e. the smallest value ofssuch that taking any Abelian group 𝓖 of orders, there exists a functionf:E(G) → 𝓖 such that the sums of edge labels at every vertex are distinct. So far it was not known ifsg(G) is finite for disconnected graphs. In the paper we present some upper bound for all graphs. Moreover we give the exact values and bounds onsg(G) for disconnected graphs without a star as a component.

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Properly even harmonious labelings of disconnected graphs

AKCE International Journal of Graphs and Combinatorics ◽

10.1016/j.akcej.2015.11.015 ◽

2015 ◽

Vol 12 (2-3) ◽

pp. 193-203 ◽

Cited By ~ 1

Author(s):

Joseph A. Gallian ◽

Danielle Stewart

Keyword(s):

Disconnected Graphs

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k-Degenerate Graphs

Canadian Journal of Mathematics ◽

10.4153/cjm-1970-125-1 ◽

1970 ◽

Vol 22 (5) ◽

pp. 1082-1096 ◽

Cited By ~ 116

Author(s):

Don R. Lick ◽

Arthur T. White

Keyword(s):

Bipartite Graph ◽

Chromatic Number ◽

Complete Graph ◽

Planar Graphs ◽

Complete Bipartite Graph ◽

Natural Numbers ◽

Image Position ◽

Disconnected Graphs ◽

Totally Disconnected

Graphs possessing a certain property are often characterized in terms of a type of configuration or subgraph which they cannot possess. For example, a graph is totally disconnected (or, has chromatic number one) if and only if it contains no lines; a graph is a forest (or, has point-arboricity one) if and only if it contains no cycles. Chartrand, Geller, and Hedetniemi [2] defined a graph to have property Pn if it contains no subgraph homeomorphic from the complete graph Kn+1 or the complete bipartite graphFor the first four natural numbers n, the graphs with property Pn are exactly the totally disconnected graphs, forests, outerplanar and planar graphs, respectively. This unification suggested the extension of many results known to hold for one of the above four classes of graphs to one or more of the remaining classes.

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