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What does a graph with six vertice and each vertex is of one degree look like?
Wiki User
∙ 12y ago
Six points.
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∙ 8y ago
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How can you understand a given graph is Euler or not?
The definition of an Eulerian path is a path in a graph which visits each edge exactly once. Intuitively, think of tracing the path with a pencil without lifting the pencil's edge from the page. One definition of an Eulerian graph is that every vertex has an even degree. You can check this by counting the degrees. Please see the related link for details.
Does a graph of a circle represent a graph of a function?
Although closely related problems in discrete geometry had been studied earlier, e.g. by Scott (1970) and Jamison (1984), the problem of determining the slope number of a graph was introduced byWade & Chu (1994), who showed that the slope number of an n-vertex complete graph Knis exactly n. A drawing with this slope number may be formed by placing the vertices of the graph on a regular polygon. As Keszegh, Pach & Pálvölgyi (2011) showed, every planar graph has a planar straight-line drawing in which the number of distinct slopes is a function of the degree of the graph. Their proof follows a construction of Malitz & Papakostas (1994) for bounding the angular resolution of planar graphs as a function of degree, by completing the graph to a maximal planar graph without increasing its degree by more than a constant factor, and applying the circle packing theorem to represent this augmented graph as a collection of tangent circles. If the degree of the initial graph is bounded, the ratio between the radii of adjacent circles in the packing will also be bounded, which in turn implies that using a quadtree to place each graph vertex on a point within its circle will produce slopes that are ratios of small integers. The number of distinct slopes produced by this construction is exponential in the degree of the graph.
What is formed at each vertex of the polygon?
An angle is formed at each vertex of a polygon.
What can you make out of 4 trapezoids?
Eight quadrilaterals if you cut each one in two from side to side (not vertex to opposite vertex).Eight quadrilaterals if you cut each one in two from side to side (not vertex to opposite vertex).Eight quadrilaterals if you cut each one in two from side to side (not vertex to opposite vertex).Eight quadrilaterals if you cut each one in two from side to side (not vertex to opposite vertex).
How many edges of the cube meet at each vertex?
Three edges meet at each vertex.
What is dense graph and sparse graph?
Sparse vs. Dense GraphsInformally, a graph with relatively few edges is sparse, and a graph with many edges is dense. The following definition defines precisely what we mean when we say that a graph ``has relatively few edges'': Definition (Sparse Graph) A sparse graph is a graph in which .For example, consider a graph with n nodes. Suppose that the out-degree of each vertex in G is some fixed constant k. Graph G is a sparse graph because .A graph that is not sparse is said to be dense:Definition (Dense Graph) A dense graph is a graph in which .For example, consider a graph with n nodes. Suppose that the out-degree of each vertex in G is some fraction fof n, . E.g., if n=16 and f=0.25, the out-degree of each node is 4. Graph G is a dense graph because .
Always use the vertex and at least points to graph each quadratic equation?
You should always use the vertex and at least two points to graph each quadratic equation. A good choice for two points are the intercepts of the quadratic equation.
When does a directed acyclic graph yield a unique topological sort?
Understanding when a Directed Acyclic Graph (DAG) yields a unique topological sort is an intriguing aspect of graph theory and algorithms. A Directed Acyclic Graph is a graph with directed edges and no cycles. Topological sorting for a DAG is a linear ordering of vertices such that for every directed edge uv from vertex u to vertex v, u comes before v in the ordering. A unique topological sort in a DAG occurs under a specific condition: when the graph has a unique way to visit its vertices without violating the edge directions. This is possible only if the graph has a unique Hamiltonian path, meaning there is a single path that visits every vertex exactly once. To determine if a DAG has a unique topological sort, you can check for the presence of a Hamiltonian path. One approach to do this is using the concept of in-degree and out-degree of vertices (the number of incoming and outgoing edges, respectively). For a DAG to have a unique topological sort, each vertex except one must have an out-degree of exactly one. Similarly, each vertex except one must have an in-degree of exactly one. The starting vertex of the Hamiltonian path will have an out-degree of one and in-degree of zero, and the ending vertex will have an out-degree of zero and in-degree of one. If these conditions are met, the DAG will have a unique topological sort. In practical applications, this concept is significant in scenarios where tasks need to be performed in a specific order. For example, in project scheduling or course prerequisite planning, knowing whether a DAG has a unique topological sort can help in determining if there is only one way to schedule tasks or plan courses. In summary, a Directed Acyclic Graph yields a unique topological sort if and only if it contains a unique Hamiltonian path. This scenario is characterized by each vertex (except for the start and end) having exactly one in-degree and one out-degree. Understanding this concept is crucial in areas like scheduling and planning, where order and precedence are key.
How can you construct a dual graph?
A dual graph is constructed by taking the original graph, which must be planar (no crossing edges) and creating a vertex inside each face of the graph. A face is an enclosed area in the graph and the space outside of the graph is also a face. Once you have created a vertex in every space, you must connect every vertex by crossing each edge in the original graph. For example, a simple triangle is planar and has two faces, one inside and one outside. We would form a vertex inside the triangle and somewhere outside of the triangle. Now, we have three edges we must cross, so starting at the inner vertex, draw three lines with one exiting through exactly 1 side each. You should now have a vertex with 3 lines that exist outside of the triangle. Without crossing them, just simply connect them to the vertex on the outside. This will create a dual of the triangle. It should resemble two vertices connected with three edges. Note that this dual graph is not planar like the original.
What is a hamiltonian path in a graph?
A path along the edges of a graph that traverses every vertex exactly once and terminates at its starting point. Also known as Hamiltonian circuit; Hamiltonian cycle.
How can you understand a given graph is Euler or not?
The definition of an Eulerian path is a path in a graph which visits each edge exactly once. Intuitively, think of tracing the path with a pencil without lifting the pencil's edge from the page. One definition of an Eulerian graph is that every vertex has an even degree. You can check this by counting the degrees. Please see the related link for details.
Does a graph of a circle represent a graph of a function?
Although closely related problems in discrete geometry had been studied earlier, e.g. by Scott (1970) and Jamison (1984), the problem of determining the slope number of a graph was introduced byWade & Chu (1994), who showed that the slope number of an n-vertex complete graph Knis exactly n. A drawing with this slope number may be formed by placing the vertices of the graph on a regular polygon. As Keszegh, Pach & Pálvölgyi (2011) showed, every planar graph has a planar straight-line drawing in which the number of distinct slopes is a function of the degree of the graph. Their proof follows a construction of Malitz & Papakostas (1994) for bounding the angular resolution of planar graphs as a function of degree, by completing the graph to a maximal planar graph without increasing its degree by more than a constant factor, and applying the circle packing theorem to represent this augmented graph as a collection of tangent circles. If the degree of the initial graph is bounded, the ratio between the radii of adjacent circles in the packing will also be bounded, which in turn implies that using a quadtree to place each graph vertex on a point within its circle will produce slopes that are ratios of small integers. The number of distinct slopes produced by this construction is exponential in the degree of the graph.
How many edges are there in a graph with 7 vertices each with degree 2?
There are 7 edges.
Design an algorithm to check if a given graph is connected?
As a rough outline, we start with some vertex x, and build a list of the vertices you can get to from x. Each time we find a new vertex to be added to this list, we check its neighbors to see if they should be added as well. Finally, we check whether the list covers the whole graph. In pseudocode: test-connected(G){choose a vertex xmake a list L of vertices reachable from x,and another list K of vertices to be explored.initially, L = K = x.while K is nonemptyfind and remove some vertex y in Kfor each edge (y,z)if (z is not in L)add z to both L and Kif L has fewer than n itemsreturn disconnectedelse return connected}To analyze the algorithm, first notice that the outer loop happens n times (once per vertex). The time for the inner loop (finding all unreached neighbors) is more complicated, and depends on the graph representation. One key step (testing whether z is in L) seems like it might be slow, but can be done quickly by keeping a bit on each vertex that says whether it's in L or not. * For the object oriented representation, each execution of the inner loop involves scanning through all m edges of the graph. So the total time for the algorithm is O(mn). * For the adjacency matrix representation, each execution of the inner loop involves looking at a single row of the matrix, in time O(n). So the total time for the algorithm is O(n^2). * In the adjacency list (or incidence list) representation, each element on each list is scanned through once. So the total time on all executions of the inner loop is the same as the total length of all adjacency lists, which is 2m. Note that we don't multiply this by n, even though this is a nested loop -- we just add up the number of times each statement is executed in the overall algorithm. The total time for the algorithm is O(m+n) At the end of the algorithm, the list L tells you one connected component of the graph (how much of the graph can be reached from x). With some more care, we can find all components of G rather than just one component. If graph is connected, we can modify the algorithm to find a tree in G covering all vertices (a spanning tree): For each z, let parent(z) be the vertex y corresponding to the time at which we added z to L. This gives a graph in which each vertex except x is connected to some previous vertex, and any such graph must be a tree
What is formed at each vertex of the polygon?
An angle is formed at each vertex of a polygon.
What can you make out of 4 trapezoids?
Eight quadrilaterals if you cut each one in two from side to side (not vertex to opposite vertex).Eight quadrilaterals if you cut each one in two from side to side (not vertex to opposite vertex).Eight quadrilaterals if you cut each one in two from side to side (not vertex to opposite vertex).Eight quadrilaterals if you cut each one in two from side to side (not vertex to opposite vertex).
What is the measure of each side of a regular hexagon?
Half the length from one vertice to its opposite.
