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In the mathematical field of graph theory the degree matrix is a diagonal matrix which contains information about the degree of each vertex.

Definition

Given a graph <math>G=(V,E)</math> with <math>\|V\|=n</math> the degree matrix <math>D</math> for <math>G</math> is a <math>n \times n</math> square matrix defined as

<math>d_{i,j}:=\left\{

\begin{matrix}
\deg(v_i) & \mbox{if}\ i = j \\
0 & \mbox{otherwise}
\end{matrix}
\right.
</math>

Example

For an undirected graph, the degree of a vertex is the number of edges incident to the vertex. This means that each loop is counted twice. This is because each edge has two endpoints and each endpoint adds to the degree.

• The degree matrix of a k-regular graph has a constant diagonal of <math>k</math>

In France, a DEA (diplôme d’études approfondies, or “diploma of advanced studies”) is a former postgraduate degree.

The DEA was a 1-2 years degree taken after a maîtrise (4-year French undergraduate program) in a university prior to entering a doctoral program. The DEA was generally the first year of a doctoral program and is sometimes referred to as a “first doctoral degree”. It is generally equivalent to the North American “All But Dissertation” or ABD status within a doctoral program or a Master of Philosophy (M.Phil.) or Master of Research (M.Res.) degree. But in the francophonic universities in Switzerland the DEA / DESS are 1-2 years degrees taken after a licence (4-year Swiss undergraduate degree).

In the hard sciences, it generally consisted of a number of courses, with examinations and grades, followed by an internship in a scientific laboratory. The students would then write a short thesis about the scientific work they did in their internship, and defend this thesis in front of a committee. The grades of the examinations and, if possible, the internship, were used to rank students in order to select those eligible for funding for doctoral studies. Students that would follow up with a Doctorat (Ph.D.) would generally do it in the same laboratory as their internship. DEA’s are now subsumed into research-oriented Master’s degrees.

• Tennessee Center for Policy Research Gore

In graph theory, for a connected graph <math>G</math>, a spanning tree <math>T</math> is a subgraph of <math>G</math> with the least number of edges that still spans <math>G</math>. A number of properties can be proved about <math>T</math>. <math>T</math> is acyclic, has (<math>|V|-1</math>) edges where <math>V</math> is the number of vertices in <math>G</math> etc.

A minimum degree spanning tree <math>T’</math> is a spanning tree which has the least degree. The vertex of maximum degree in <math>T’</math> is the least among all possible spanning trees of <math>G</math>.

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The degree of polymerization, or DP, is the number of repeat units in an average polymer chain at time t in a polymerization reaction <ref> Gold Book definition Link</ref>. The length is in monomer units. The degree of polymerization is a measure of molecular weight. For most industrial purposes, lengths in the thousands or tens of thousands are desired.

For a homopolymer, the degree of polymerization may be calculated as shown bellow

DP = Total MW of the polymer/MW of the repeating unit

<math>M_n=X_nM_o</math>

In polycondensation, in order to achieve a high degree of polymerization (and hence molecular weight), X_n, a high fractional monomer conversion, p, is required, as per Carothers’ equation: X_n = 1/(1−p). A monomer conversion of p = 99% would be required to achieve X_n = 100.

Correlation with physical properties

Polymers with identical composition but different total molecular weights may exhibit different physical properties. In general, increasing degree of polymerization correlates with higher melting temperature <ref> Flory, P.J. and Vrij, A. J. Am. Chem. Soc.; 1963; 85(22) pp3548-3553 </ref> and higher mechanical strength.

References

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