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A simple graph is a fundamental concept in graph theory. It is defined as a type of graph where each pair of vertices is connected by at most one edge, and no edge starts and ends at the same vertex—essentially, there are no loops or parallel edges. Simple graphs are widely used because they are easier to analyze and conceptualize compared to more complex graphs that may have multiple edges between two vertices or self-connections.

In a simple graph, the absence of loops and multiple edges means that its edge set can be entirely defined by a simple list of pairs of vertices. This property makes simple graphs massively appealing for solving problems such as connectivity, pathfinding, and network flow. Analyses involving simple graphs often give insights into relationships and structures in fields ranging from chemistry to computer science.

The degree of a vertex in a graph is a crucial concept as it denotes the number of edges incident to that vertex. In a simple graph with no loops or parallel edges, this can be simply counted as the number of direct connections. To put it in another way, it's the number of edges connecting to a vertex from its neighboring vertices.

For example, if a vertex is connected to three other vertices by edges, its degree is 3. Note that in a simple graph with a total of \(N\) vertices, the degree of any vertex can range from 0 to \(N-1\). The maximum degree \(N-1\) in a simple graph signifies that a vertex is directly connected to every other vertex in the graph.

Understanding vertex degree is vital because it helps in determining the graph's properties such as connectivity, the possibility of Eulerian paths or circuits, and more. In the problem above, by applying the pigeonhole principle or observing logical contradictions, it is shown that a simple graph must have at least two vertices sharing the same degree.

Graph properties refer to the characteristics and qualities that define different types of graphs. When analyzing graphs, several properties are often considered:

• Degree Sequence: It’s the list of vertex degrees of a graph. The degree sequence can help in understanding the distribution of connections across the graph.
• Connectedness: This property determines whether there is a path between every pair of vertices. A simple graph being connected means there are no isolated vertices.
• Cyclic: A graph is cyclic if it contains at least one cycle, meaning it is possible to begin at a vertex, travel along a path, and return to the original vertex without retracing any edge.

Graph properties such as these are essential for discerning and predicting behaviors of networks and systems modeled as graphs. In our original exercise, the property being investigated was the degree distribution among vertices, showcasing that no matter the number of vertices \(N\), a simple graph cannot showcase entirely distinct vertex degrees for each vertex due to logical restrictions, specifically explained by the pigeonhole principle.