Laplacian spectra of power graphs of finite cyclic groups, dicyclic groups and p-groups are investigated. In the thesis we study the Laplacian spectra of power graphs Zn, Qn and p-groups.
Numbers and sets
Graphs
A separating set of Γ with the least cardinality is called a minimal separating set of Γ. The vertex connectivity of a graph Γ, denoted by κpΓq, is the minimum number of vertices whose removal results in a disconnected or trivial graph. A decoupling set of Γ is a set of edges whose removal increases the number of components of Γ.
Groups
For any integer n ě 3, the dihedral group Dn is a finite group of order 2n with presentation. where e is the identity element of Dn. GroupDn fulfills the following properties. For any integer n ě 2, the dicyclic group Qn is a finite group of order 4n with presentation. where e is the identity element of Qn.
Laplacian spectra
For any graphΓ, the multiplicity of 0 as an eigenvalue of LpΓq is equal to the number of components ofΓ.
Power graphs
In view of Remark 1.5.5 and since every cyclic group of order n is isomorphic to Zn, we study it for Zn instead of studying the graph-theoretic properties of the power graph of an arbitrary finite cyclic group of order n. i) The power graph GpGq is connected. ii) The power graph GpGq is complete if and only if G is a cyclic group of order 1 or pα, for a given prime number p and αP N. iii). The following theorem establishes that the Laplace spectral power radius of any finite group is its order.
Minimal separating sets
ThenT is a minimally separable set of GpGq if and only if rTs is a minimally separable set of GpGq.r. Since T is a disjoint set of GpGq, there exists S ĎT such that S is a minimal disjoint set of GpGq.
Vertex connectivity of GpZ n q
If n ±1 is not a prime number, the following statement holds. i) If n is not a prime power, then every disjoint set of GpZnq contains SpZnq. For nPN the following statements are equivalent. i) n is a product of two different prime numbers. ii) SpZnq is a disjoint set of GpZnq. Assume that n P N is neither a prime power nor a product of two different primes. i) N1prasq is a segregating set of G1pZnq.
Then for each 1ďk ďr, the following statements hold. is a minimal separating group of G1pZnq. is not a minimal separating set of G1pZnq. Thus T is a minimal separating group of Gr1pZnq and hence the proof follows from Lemma 2.3.12(iii).
Conclusion
In Section 3.1 we show that in every component of a regular p-group power graph, elements of order are adjacent to all other elements. Using this, we find the number of components of the proper power graph of the abelian p-group. If C is one such component, we show that C has exactly p´1 elements of order and that all other vertices in C are adjacent to them.
We then calculate the number of components of the correct power graph of an abelian p-group. If xPG˚ is an element of order p, then x is adjacent to every other vertex of Cpxq.
Structure
It follows from Theorem 3.1.5 that the real power graph of an acyclic abelian p-group has more than one component. For ease of notation, for each pa, bq P Zpα ˆZpβ we denote Γppa, bqq simply by Γpa, bq. In the rest of this section, the basic group of presented results is Zpα ˆZpβ.
This implies|b, which contradicts the fact that gcdpb, pq “ 1. So rpapk, bqs has no primitive class. By induction hypothesis, Γppα´1, pmq –Γp0, pmq, so that. 3.14) Since km`1´1ěm it follows from the induction hypothesis that.
Conclusion
In this chapter we study the minimum degree of power graphs of finite groups and determine its relation to connectivity. In Section 4.1 we establish that the edge connectivity and the minimum degree of power graphs of finite groups are equal. Following from this, we determine the minimum degree of power graphs of abelian p-groups,Dn.
Along with minimum degree, we also obtain minimal disconnection sets of power graphs for these groups. In Section 4.3, we characterize the similarity of vertex connectivity and minimum degree of power graphs for Zn and abelian p-groups.
Edge connectivity
Minimum degree
Finite cyclic group
If n P N is a composite number, then there exists a proper divisor cą1 of n such that δpGpZnqq “degpcq. Thus there exists a P Z1n such that δpGpZnqq “degpaq. i) If it is a composite number, then δpGpZnqq “φpnq`1`δpG1pZnqq. i) Let n PN be a composite number. Then by Proposition 2.3.3, G1pZnq breaks down and its component induced by xpy˚ has p as its only vertex.
It is easy to see that the subgraphs induced by xpy˚ and xqy˚ are the only components of G1pZnq. According to Lemma 4.2.3, to determine δpGpZnqq, it is enough to compare the degrees of the vertices of the form c, where c±1 is a real divisor of n.
Abelian p-group
If g P G such that all components of ψpgq are 0 except tth, say a, satisfying gcdpa, pq “ 1, then Erg, ψ´1pxψpgqyq ´gs is a minimal disjoint set GpGq.
Dihedral and dicyclic group
Equality of vertex connectivity and minimum degree
If G is not a cyclic group of prime-power order and δpGpGqq “degpgq for some g PG, then the following statements hold. i) Npgq is the minimal separating set of GpGq. Therefore, we conclude that Npgq is a minimal separating set of GpGq. ii) By (i), Npgq is the minimal separating set of GpGq. As a consequence, we give the minimal separating set and the minimal separating set GpZnq.
Let a be the vertex such that degpaq “ k. Assume that the integer n ±1 is not a prime power. race, e.g. A, is a minimum separation set and E“n. ThenκpGpGqq “ δpGpGqqif and only if σpGq “1 or τpGq “ 2. ThusκpGpGqq “δpGpGqq if and only if τpGq “2. Whereas, when σpGq ± 1 andτpGq ±2, then κpGpGqq ‰δpGpGqq.
Conclusion
In particular, Laplacian spectra of power graphs of Zn and Dn were studied by Chattopadhyay and Panigrahi [2015]. In this chapter we study Laplacian spectra of power graphs of Zn, Qn and p-groups in Sections 5.2, 5.3 and 5.4, respectively. 2002] investigated the equality of vertex connectivity and algebraic connectivity of graphs that are not complete and connected. We then obtain some results that are essential for the study of Laplacian spectrum of power graphs.
Applying Theorem 5.1.1 and Theorem 5.1.2, we now give alternative and shorter proofs for some results on the Laplacian spectra of power graphs of Zn, Dn, and Qn from [Chattopadhyay, 2015; Chattopadhyay and Panigrahi, 2015]. We now present some results that are fundamental to Laplacian spectra of power graphs and valid for all finite groups.
Finite cyclic group
We further show in Theorem 5.2.3 that for equality (5.5) to hold, the aforementioned sufficient condition is also necessary. For an integer n ±1, the multiplicity of the Laplacian eigenvalue n of GpZnq is φpnq `1 if and only if n"4 or n has at least two prime factors. For an integerą1, the algebraic connection of GpZnqisφpnq`1 if and only if n is a prime or a product of two different primes.
If n is a prime number or a product of two different prime numbers, then the algebraic connectivity is λn´1pGpZnqq φpnq `1 (cf. Theorem 1.5.27). For an integer ną1, κpGpZnqq “λn´1pGpZnqqif holds and only if n is the product of two different prime numbers.
Dicyclic group
Accordingly, by Theorem 1.4.9, the multiplicity of 0 as a Laplace eigenvalue of. GpQnq is three if n is a power of 2, and two otherwise. By Theorem 1.4.10, the multiplicity of 4n as a Laplace eigenvalue of GpQnq is equal to one less than the multiplicity of 0 as a Laplace eigenvalue of GpQnq. If Qn is a generalized quaternion, it follows from Theorem 5.1.6 that GpQnq is a Laplace integral.
Applying Theorem 1.4.11, we thus conclude that the Laplacian eigenvalues of G˚pGq above are bounded by n´p. Then it follows from Corollary 1.5.28 and Theorem 5.1.6 that the multiplicity of n is at least two. If G is neither a cyclic nor a generalized quaternion, it follows from Proposition 5.1.9(i) and Theorem 5.4.2 that κpGpGqq “1“λn´1pGpGqq.
In the next theorem we apply Theorem 5.4.9 to determine the lower bound of the multiplicity of pk, kP Nas a Laplace eigenvalue of the power graph of a p-group.
Conclusion
Vertex connectivity
In this subsection, we show that GpZnq is not critically vertex connected when n is a product of two or three distinct primes. Finally, we provide a necessary and sufficient condition for the power graph of a p-group to be critically vertex connected. If n is a product of two or three distinct primes, then GpZnq is not critically vertex connected.
Following the proof of Theorem 2.3.22, a minimal separation set of GpGpq is exactly the union of any two «p. If G is a p-group, then GpGq is critically vertex connected if and only if G is cyclic.
Edge connectivity
If G is a p-group, then GpGq is critically edge-connected if and only if G is cyclic or G“Q2.
Minimal connectivity
Vertex connectivity
We then apply these results to power graphs and obtain a characterization such that the power graph of a finite group is minimally vertex connected. If Γ is a graph with minimal vertex, then for any edge ε, no minimal separation set of Γ´ε contains endpoints of ε. If Γ´ε is connected and S is a minimal separating set of Γ´ε, then SY tuu or SY tvu is a minimal separating set of Γ.
For a finite group G, GpGq is minimal vertex connected if and only if G is a cyclic group of prime power order or G is an elementary abelian 2-group. For any p-group G, GpGq is minimal vertex connected if and only if G is a cyclic or an elementary abelian 2-group.
Edge connectivity
While, if G is an elementary abelian 2-group, κpGpGqq “1 and GpGq ´ε are decoupled for each edgeε in GpGq. Then GpGq is minimally edge connected and κ1pGpGqq “1 if and only if G is an elementary abelian 2-group. Then GpGq is minimally edge-connected if and only if G is a cyclic group with primary power order or if exppGq is prime.
For any odd integer n ±0, GpZnq is minimally connected to the edge if and only if n is a prime power. Then GpGq is minimally edge connected if and only if G is a cyclic or or exppGq “p.
Conclusion
Chattopadhyay, S.: 2015, Some graph-theoretic and spectral results on effect graphs for certain finite groups, PhD thesis, IIT Kharagpur, India. J.: 2000, A combinatorial property and power graph for groups, Contributions to General Algebra, 12 (Vienna, 1999), Heyn, Klagenfurt, pp.
