### Counting the Number of Disconnected Labeled Graphs of Order Five without Paralel Edges

#### Abstract

** Abstract**— Given a graph G(V,E) with n vertices and m edges, where every vertex is labeled, there are a lot of possible graphs that can be constructed, either connected graphs or disconnected, simple or not simple. A graph G(V,E) is called as a connected graph if there exists at least one path between every pair of vertices in G, and otherwise, G is disconnected. A graph G is called as a labeled graph if every node/vertex and or every edge is labeled. In this research, we are concerning about a graph where every vertex is labeled. Parallel edges are two edges or more which have the same end points. In this research we found that the number of disconnected labeled graph without parallel edges for and can be obtained with the following formula:

{{. is the number of disconnected labeled graph without parallel edges for and .

** Keywords**— counting graph, labeled graph, disconnected, parallel edges

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DOI: http://dx.doi.org/10.23960/ins.v1i1.7

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