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Page "Polynomial" ¶ 63
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graph and polynomial
The questions range from counting ( e. g., the number of graphs on n vertices with k edges ) to structural ( e. g., which graphs contain Hamiltonian cycles ) to algebraic questions ( e. g., given a graph G and two numbers x and y, does the Tutte polynomial T < sub > G </ sub >( x, y ) have a combinatorial interpretation ?).
0 is the polynomial equation corresponding to P. The solutions of this equation are called the roots of the polynomial ; they are the zeroes of the function ƒ ( corresponding to the points where the graph of ƒ meets the x-axis ).
A polynomial function in one real variable can be represented by a graph.
* The graph of the zero polynomial
* The graph of a degree 0 polynomial
* The graph of a degree 1 polynomial ( or linear function )
* The graph of a degree 2 polynomial
* The graph of a degree 3 polynomial
* The graph of any polynomial with degree 2 or greater
so the graph of any function which is a polynomial of degree 2 in x is a parabola with a vertical axis.
" can be solved in polynomial time, and in fact, for a graph with V vertices and E edges, it can be solved in O ( VE ) time.
In a different usage to the above, a polynomial of degree 1 is said to be linear, because the graph of a function of that form is a line.
While it is NP-hard to determine the treewidth of a graph, many NP-hard combinatorial problems on graphs are solvable in polynomial time when restricted to graphs of bounded treewidth.
* has characteristic polynomial, making it an integral grapha graph whose spectrum consists entirely of integers.
The number of acyclic orientations is equal to | χ (- 1 )|, where χ is the chromatic polynomial of the given graph.
The characteristic polynomial of a graph is the characteristic polynomial of its adjacency matrix.
It is a graph invariant, though it is not complete: the smallest pair of non-isomorphic graphs with the same characteristic polynomial have five nodes.
Two-dimensional graph of a cubic, the polynomial ƒ ( x ) = 2x < sup > 3 </ sup > − 3x < sup > 2 </ sup > − 3x + 2.
Their work has recently been followed by a flood of papers — though not as many as on the Tutte polynomial of a graph.
The problem is fixed-parameter tractable, meaning that there is an algorithm whose running time can be bounded by a polynomial function of the size of the graph multiplied by a larger function of.
It is also the index of the first nonzero coefficient of the chromatic polynomial of a graph.

graph and always
For a Lipschitz continuous function, there is a double cone ( shown in white ) whose vertex can be translated along the graph, so that the graph always remains entirely outside the cone.
In general, for a puzzle with n disks, there are 3 < sup > n </ sup > nodes in the graph ; every node has three edges to other nodes, except the three corner nodes, which have two: it is always possible to move the smallest disk to the one of the two other pegs ; and it is possible to move one disk between those two pegs except in the situation where all disks are stacked on one peg.
Note that although a function is always identified with its graph, they are not the same because it will happen that two functions with different codomain could have the same graph.
A recursive operator is an enumeration operator that, when given the graph of a partial recursive function, always returns the graph of a partial recursive function.
We represent the surface by the implicit function theorem as the graph of a function, f, of two variables, in such a way that the point p is a critical point, i. e., the gradient of f vanishes ( this can always be attained by a suitable rigid motion ).
A bit of thought shows that this process always changes the parity of the number of rows, and applying the process twice brings us back to the original graph.
This point of view has the advantage that edge deletions leave the rank of a graph unchanged, and edge contractions always reduce the rank by one.
In this variation of graph minor theory, a graph is always simplified after any edge contraction to eliminate its self-loops and multiple edges .< ref > is inconsistent about whether to allow self-loops and multiple adjacencies: he writes on p. 76 that " parallel edges and loops are allowed " but on p. 77 he states that " a graph is a forest if and only if it does not contain the triangle K < sub > 3 </ sub > as a minor ", true only for simple graphs .</ ref >
Hill climbing will follow the graph from vertex to vertex, always locally increasing ( or decreasing ) the value of, until a local maximum ( or local minimum ) is reached.
The Hadwiger number of a graph is the number of vertices in the largest clique that can be formed as a minor in the graph ; the Hadwiger conjecture states that this is always at least as large as the chromatic number.
A k-vertex-connected graph is a graph in which removing fewer than k vertices always leaves the remaining graph connected.
A source is bound to exist because the graph is a directed acyclic graph by construction, and such graphs always have sources.
Put another way, Whitney's theorem guarantees that the line graph almost always encodes the topology of the original graph G faithfully but it does not guarantee that dynamics on these two graphs have a simple relationship.
But in graph theory, when the term is used without any qualification, it almost always refers to the chromatic number of a graph.

graph and tends
For curves given by the graph of a function, horizontal asymptotes are horizontal lines that the graph of the function approaches as x tends to Vertical asymptotes are vertical lines near which the function grows without bound.
Horizontal asymptotes are horizontal lines that the graph of the function approaches as x tends to +∞ or −∞.
Hypergraph theory tends to concern questions similar to those of graph theory, such as connectivity and colorability, while the theory of set systems tends to ask non-graph-theoretical questions, such as those of Sperner theory.
If a cubic graph is chosen uniformly at random among all n-vertex cubic graphs, then it is very likely to be Hamiltonian: the proportion of the n-vertex cubic graphs that are Hamiltonian tends to one in the limit as n goes to infinity.
Several important graph optimization problems are APX hard, meaning that, although they have approximation algorithms whose approximation ratio is bounded by a constant, they do not have polynomial time approximation schemes whose approximation ratio tends to 1 unless P = NP.
Malkiel then took the results in a chart and graph form to a chartist, a person who “ seeks to predict future movements by seeking to interpret past patterns on the assumption that ‘ history tends to repeat itself ’”.

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