# Raziskovalni matematični seminar - Arhiv

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**30.05.2011 10:00 Seminarska soba v Galebu**

**Lecturer:**

**Alexander**

**Vasilyev**

**Title:**

**Communicability Graph and Community Structures in Complex Networks**

**Abstract:**Estrada and Hatano proposed an algorithm for the detection of community structure in complex networks using the concept of network communicability.The communities are defined as the cliques of a “communicability graph”, which has the same set of nodes as the complex network and links determined by the communicability function. Then, the problem of finding the network communities is transformed to an all-clique problem of the communicability graph. In addition, we extend here the concept of the communicability to account for the strength of the interactions between the nodes by using the concept of inverse temperature of the network. We develop an algorithm to manage the different degrees of overlapping between the communities in a complex network. Finally, we amend this algorithm by eliminating the subjectivity of choosing degree of overlapping and including an additional check of the fitness of detected communities. We show that this amendment can detect some communities which remain undetected by the original Estrada and Hatano’s algorithm.

**SLIDES FROM THE TALK ARE AVAILABLE HERE: DOWNLOAD!**

**23.05.2011 10:00 Seminarska soba v Galebu**

**Lecturer: Anna Klymenko**

**Title: The Coxeter Graph**

**Abstract:**

**According to Lovasz conjecture, every finite connected vertex-transitive graph contains a Hamiltonian cycle except**

**Slides from the talk are available here: DOWNLOAD!**

**16.05.2011 10:00 Seminarska soba v Galebu**

**Title: Median graphs and algebras**

**Abstract**: In this talk we present a strong connection between certain ternary algebras, called median algebras, that enjoy three simple axioms, and the class of median graphs. The latter are defined as the graphs in which for any triple of vertices the three intervals between pairs of the triple share exactly one vertex. Median graphs are the most important class in metric graph theory, and appear in several applications in Computer science, mathematical biology and elsewhere.

We select a small portion of numerous characterizations and properties of median graphs, with an emphasis on some recent results.

**You can download the slides from the talk here: DOWNLOAD!**

**16.05.2011 9:00 Seminarska soba v Galebu**

**Title**:

**Steiner index of a graph**

**Abstract**: Given a subset

**09.05.2011**

**10:00 Seminarska soba v Galebu**

**Title: The isomorphism classes of all generalized petersen graphs**

**Abstract**:The generalized Petersen graphs GP(

*n, k*) were introduced by Watkins in 1969, and since then have been studied extensively. For an ordered pair (

*n, k*), where 1

*≤*

*k*

*≤*

*n*

*−*1,

*k*=

*n/*2, the generalized Petersen graph GP(

*n, k*) has vertex set Z

_{2}

*×*Z

_{n}and edge set

*{*(1

*, i*)(1

*, i*+1)

*,*(0

*, ki*)(0

*, ki*+

*k*)

*,*(0

*, i*)(1

*, i*) :

*i*

*∈*Z

_{n}

*}*. Many well-known graphs are generalized Petersen graphs, with GP(5

*,*2) being the Petersen graph itself, GP(4

*,*1) being the skeleton of the three dimensional cube, GP(10

*,*2) being the skeleton of the dodecahedron, and GP(10

*,*3) is the Desargues graph (the Levi or incidence graph of the Desargues configuration). In this talk, we determine necessary and sufficient conditions for two generalized Petersen graphs GP(

*n, k*) and GP(

*n, ℓ*) to be isomorphic. The conditions are that either

*k*

*≡ ±ℓ*(mod

*n*) or that

*kℓ*

*≡ ±*1 (mod

*n*).

**You can download the slides from the talk here: DOWNLOAD!**