*4.3. Spectral Clustering Results*

Regarding to the results obtained by the *k*-means and GMM algorithms, these have a certain relationship, since it turns out that both are consistent with each other; however, the results of the spectral grouping (see Table 4 and Figure 10) are complementary to the two techniques already mentioned and analyzed, this is because the spectral grouping emphasizes the changes in the less favored universities; where it is shown that some universities in the first quadrant want to improve, either in higher learning or in research, although their efforts are more modest.

#### 4.3.1. Universities in Transition in Spectral Clustering Analysis

Regarding the place where the institutions with good results in research and not so favorable results in higher learning reside, there are the College of Postgraduates (COLPOS) and Autonomous Technological Institute of Mexico (ITAM), which appear from 2009 to 2015 and in 2017; the College of México (COLMEX) that appeared from 2009 to 2012 and from 2014 to 2015; the Autonomous University of Campeche (UACAM) that is presented from 2009 to 2011 and in 2015; the UACM from 2009 to 2011 and from 2014 to 2015; the University of the Americas Puebla (UDLAP) from 2009 to 2012 and in 2014 and Chapingo Autonomous University (CHAPINGO) from 2010 to 2011, from 2014 to 2015 and in 2017. Likewise, the University of Colima (UCOL) only appears twice, in 2015 and 2017 and at the Autonomous University of Yucatán (UAY) only entered in 2015.

A particularity of this area occurs in 2016, when the group with minor higher learning and major research had no members.

For the sector with the most outstanding achievements in higher learning rather than in research, the institutions UPN, UVM, LASALLE, Autonomous University Benito Juarez of Oaxaca (UABJO), Autonomous University of Guadalajara (UAG), Autonomous University of Coahuila (UAdeC), Autonomous University of Chiapas (UANCH) and UNITEC ge<sup>t</sup> this place every year, Anahuac University (ANAHUAC) does from 2009 to 2013 and from 2015 to 2016; the University of the Mexican Army and Air force (UDEFA) and the Juarez Autonomous University of Tabasco (UJAT) shown in this place in the years 2009 to 2014 and 2016; the Technological Institute of Sonora (ITSON) from 2009 to 2014; the Autonomous University of Chihuahua (UACH) in the years 2009 to 2013; the Juarez University of Durango State (UJED) from 2009 to 2014 and from 2016 to 2017. The Popular Autonomous University of Puebla State (UPAEP) in the corresponding years from 2010 to 2017; the Autonomous University of Nayarit (UAN) and the UP in the years 2010 to 2014 and 2016 to 2017; the University of Monterrey (UDEM) in the years 2013 to 2017 and the Autonomous University of Tlaxcala (UATX) in the years 2013 to 2014 and 2016 to 2017. There are some other institutions which were not considered because they were not repeated more than three times in the entire period from 2009 to 2017.


**Table 4.** Results from 2009 to 2017 applying Spectral clustering.


*Mathematics* **2021**, *9*, 1615

**Table 4.** *Cont.*

UQROO

4.3.2. Static Universities in Spectral Clustering Analysis

Finally, the results obtained by the spectral classifier with respect to disadvantaged institutions both in research and higher learning aspects. show that the Autonomous Agrarian University Antonio Narro (UAAAN), Autonomous University of Baja California Sur (UABCS), Regiomontana University (UERRE), Autonomous University del Carmen (UNACAR), UNINTER and the University of Querétaro (UQROO) have this site all years; Western Institute of Technology and Higher Studies (ITESO) does so in the period from 2009 to 2016; CHAPINGO in the years 2009, 2012, 2013 and 2016; UATX and UDEM in the period from 2009 to 2011, but reappeared in 2015 and in 2013. The UACAM from 2012 to 2017 and the UDLAP from 2013 and during the period from 2015 to 2017. Although there are other institutions that occupy this place, these were not mentioned because their presence was not repeated for at least four years.
