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IBS Colloquium (Dec. 04)

**Center for Molecular Spectroscopy and Dynamics**

COLLOQUIUM | |

q **SPEAKER**

**Dr. Kinkar Chandra Das**

**Department of Mathematics, Sungkyunkwan University**

q **TITLE**

**Spectral graph theory: Four common spectra**

q **ABSTRACT**

Spectral graph theory studies connections between combinatorial properties of graphs

and the eigenvalues of matrices associated to the graph, such as the adjacency matrix, the

Laplacian matrix, the signless Laplacian matrix and the normalized Laplacian matrix. Let

*G *= (*V; E*) be a simple graph. Denote by *D*(*G*) the diagonal matrix of its vertex degrees

and by *A*(*G*) its adjacency matrix. Then the Laplacian matrix of *G *is *L*(*G*) = *D*(*G*) *����*

*A*(*G*). Denote the spectrum of *L*(*G*) by (*L*(*G*)) = (*_*1*; _*2*; : : : ; _**n*), where we assume the

eigenvalues to be arranged in nonincreasing order: *_*1 *_ _*2 *_ _ _ _ _ _**n**����*1 *_ _**n *= 0. Let

*a *be the algebraic connectivity of graph *G*. Then *a *= *_**n**����*1. Among all eigenvalues of the

Laplacian matrix of a graph, the most studied is the second smallest, called the algebraic

connectivity (a(G)) of a graph [5]. In this talk we will introduce the basics of spectral graph

theory, we will show some results on *_*1(*G*) and *a*(*G*) of graph *G*. We obtain some integer and

real Laplacian (adjacency, signless Laplacian, normalized) eigenvalues of graphs. Moreover,

we discuss several relations between eigenvalues (adjacency, Laplacian, signless Laplacian) and graph

parameters.Finally, we give some conjectures on the spectral graph theory.

q **DATE AND VENUE**

**Dec. 04, 2015 (1:00-2:00 p.m.) **

**Seminar room 119, College of Science (****아산이학관**** 119)**