Urschel introduced a notion of nodal partitioning to prove an upper bound on the number of nodal decomposition of discrete Laplacian eigenvectors. The result is an analogue to the well-known ...Courant's nodal domain theorem on continuous Laplacian. In this article, using the same notion of partitioning, we discuss the lower bound (or lack thereof) on the number of nodal decomposition of eigenvectors in the class of all graphs with a fixed number of vertices (however large). This can be treated as a discrete analogue to the results of Stern and Lewy in the continuous Laplacian case.
This paper provides a bridge between two active areas of research, the spectrum (set of element orders) and the power graph of a finite group. The order sequence of a finite group \(G\) is the list ...of orders of elements of the group, arranged in non-decreasing order. Order sequences of groups of order \(n\) are ordered by elementwise domination, forming a partially ordered set. We prove a number of results about this poset, among them the following. M.~Amiri recently proved that the poset has a unique maximal element, corresponding to the cyclic group. We show that the product of orders in a cyclic group of order \(n\) is at least \(q^{\phi(n)}\) times as large as the product in any non-cyclic group,where \(q\) is the smallest prime divisor of \(n\) and \(\phi\) is Euler's function, with a similar result for the sum. The poset of order sequences of abelian groups of order \(p^n\) is naturally isomorphic to the (well-studied) poset of partitions of \(n\) with its natural partial order. If there exists a non-nilpotent group of order \(n\), then there exists such a group whose order sequence is dominated by the order sequence of any nilpotent group of order \(n\). There is a product operation on finite ordered sequences, defined by forming all products and sorting them into non-decreasing order. The product of order sequences of groups \(G\) and \(H\) is the order sequence of a group if and only if \(|G|\) and \(|H|\) are coprime. The paper concludes with a number of open problems.
For a finite group \(G\), let \(\psi(G)\) denote the sum of element orders of \(G\). This function was introduced by Amiri, Amiri, and Isaacs in 2009 and they proved that for any finite group \(G\) ...of order \(n\), \(\psi(G)\) is maximum if and only if \(G \simeq \mathbb{Z}_n\) where \(\mathbb{Z}_n\) denotes the cyclic group of order \(n\). Furthermore, Herzog, Longobardi, and Maj in 2018 proved that if \(G\) is non-cyclic, \(\psi(G) \leq \frac{7}{11} \psi(\mathbb{Z}_n)\). Amiri and Amiri in 2014 introduced the function \(\psi_k(G)\) which is defined as the sum of the \(k\)-th powers of element orders of \(G\) and they showed that for every positive integer \(k\), \(\psi_k(G)\) is also maximum if and only if \(G\) is cyclic. In this paper, we have been able to prove that if \(G\) is a non-cyclic group of order \(n\), then \(\psi_k(G) \leq \frac{1+3.2^k}{1+2.4^k+2^k} \psi_k(\mathbb{Z}_n)\). Setting \(k=1\) in our result, we immediately get the result of Herzog et al. as a simple corollary. Besides, a recursive formula for \(\psi_k(G)\) is also obtained for finite abelian \(p\)-groups \(G\), using which one can explicitly find out the exact value of \(\psi_k(G)\) for finite abelian groups \(G\).
The classical Eulerian Numbers \(A_{n,k}\) are known to be log-concave. Let \(P_{n,k}\) and \(Q_{n,k}\) be the number of even and odd permutations with \(k\) excedances. In this paper, we show that ...\(P_{n,k}\) and \(Q_{n,k}\) are log-concave. For this, we introduce the notion of strong synchronisation and ratio-alternating which are motivated by the notion of synchronisation and ratio-dominance, introduced by Gross, Mansour, Tucker and Wang in 2014. We show similar results for Type B Coxeter Groups. We finish with some conjectures to emphasize the following: though strong synchronisation is stronger than log-concavity, many pairs of interesting combinatorial families of sequences seem to satisfy this property.
The determining number of a graph \(G = (V,E)\) is the minimum cardinality of a set \(S\subseteq V\) such that pointwise stabilizer of \(S\) under the action of \(Aut(G)\) is trivial. In this paper, ...we provide some improved upper and lower bounds on the determining number of Kneser graphs. Moreover, we provide the exact value of the determining number for some subfamilies of Kneser graphs.
The enhanced power graph of a group \(G\) is the graph \(\mathcal{G}_E(G)\) with vertex set \(G\) and edge set \( \{(u,v): u, v \in \langle w \rangle,~\mbox{for some}~ w \in G\}\). In this paper, we ...compute the spectrum of the distance matrix of the enhanced power graph of non-abelian groups of order \(pq\), dihedral groups, dicyclic groups, elementary abelian groups \(\mathrm{El}(p^n)\) and the non-cyclic abelian groups \(\mathrm{El}(p^n)\times\mathrm{El}(q^m)\) and \(\mathrm{El}(p^n)\times \mathbb{Z}_m\), where \(p\) and \(q\) are distinct primes. For the non-cyclic abelian group \(\mathrm{El}(p^n)\times \mathrm{El}(q^m)\), we also compute the spectrum of the adjacency matrix of its enhanced power graph and the spectrum of the adjacency and the distance matrix of its power graph.
For a group \(G,\) the enhanced power graph of \(G\) is a graph with vertex set \(G\) in which two distinct elements \(x, y\) are adjacent if and only if there exists an element \(w\) in \(G\) such ...that both \(x\) and \(y\) are powers of \(w.\) The proper enhanced power graph is the induced subgraph of the enhanced power graph on the set \(G \setminus S,\) where \(S\) is the set of dominating vertices of the enhanced power graph. In this paper, we first characterize the dominating vertices of enhanced power graph of any finite nilpotent group. Thereafter, we classify all nilpotent groups \(G\) such that the proper enhanced power graphs are connected and find out their diameter. We also explicitly find out the domination number of proper enhanced power graphs of finite nilpotent groups. Finally, we determine the multiplicity of the Laplacian spectral radius of the enhanced power graphs of nilpotent groups.
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