313 research outputs found
The maximal energy of classes of integral circulant graphs
The energy of a graph is the sum of the moduli of the eigenvalues of its
adjacency matrix. We study the energy of integral circulant graphs, also called
gcd graphs, which can be characterized by their vertex count and a set
of divisors of in such a way that they have vertex set
and edge set . For a fixed prime power and a fixed divisor set size , we analyze the maximal energy among all matching integral circulant
graphs. Let be the elements of .
It turns out that the differences between the exponents of
an energy maximal divisor set must satisfy certain balance conditions: (i)
either all equal , or at most the two differences
and may occur; %(for a certain depending on and ) (ii)
there are rules governing the sequence of consecutive
differences. For particular choices of and these conditions already
guarantee maximal energy and its value can be computed explicitly.Comment: Discrete Applied Mathematics (2012
Integral circulant graphs of prime power order with maximal energy
The energy of a graph is the sum of the moduli of the eigenvalues of its
adjacency matrix. We study the energy of integral circulant graphs, also called
gcd graphs, which can be characterized by their vertex count n and a set D of
divisors of n in such a way that they have vertex set Zn and edge set {{a, b} :
a, b in Zn; gcd(a - b, n) in D}. Using tools from convex optimization, we study
the maximal energy among all integral circulant graphs of prime power order ps
and varying divisor sets D. Our main result states that this maximal energy
approximately lies between s(p - 1)p^(s-1) and twice this value. We construct
suitable divisor sets for which the energy lies in this interval. We also
characterize hyperenergetic integral circulant graphs of prime power order and
exhibit an interesting topological property of their divisor sets.Comment: 25 page
The exact maximal energy of integral circulant graphs with prime power order
The energy of a graph was introduced by {\sc Gutman} in 1978 as the sum of the absolute values of the eigenvalues of its adjacency matrix. We study the energy of integral circulant graphs, also called gcd graphs,which can be characterized by their vertex count and a set of divisors of in such a way that they have vertex set and edge set .Given an arbitrary prime power , we determine all divisor sets maximising the energy of an integral circulant graph of order . This enables us to compute the maximal energy \Emax{p^s} among all integral circulant graphs of order
- …