Degree-based energies of commuting graph for dihedral groups

Graph energy is defined to be the p-norm of adjacency matrix associated to the graph for p = 1 elaborated as the sum of the absolute eigenvalues of adjacency matrix. The graph’s spectral radius represents the adjacency matrix’s largest absolute eigenvalue. Applications for graph energies and spectral radii can be found in both molecular computing and computer science. On similar lines, Inverse Sum Indeg, (ISI) energies, and (ISI) spectral radii can be constructed. This article’s main focus is the ISI energies, and ISI spectral radii of the generalized splitting and shadow graphs constructed on any regular graph. These graphs can be representation of many physical models like networks, molecules and macromolecules, chains or channels. We actually compute the relations about the ISI energies and ISI spectral radii of the newly created graphs to those of the original graph.

Let <em>G </em>= (<em>V</em>,<em>E</em>) be a simple connected graph with the vertex set <em>V</em> = {1,2,...,<em>n</em>} and sequence of vertex degrees (<em>d</em>₁,<em>d</em>₂,...,<em>d_n</em>) where <em>d_i</em> denotes the degree of a vertex <em>i</em> ∈ <em>V</em>. With <em>i</em> ∼ <em>j</em>, we denote the adjacency of the vertices <em>i</em> and <em>j</em> in the graph <em>G</em>. The inverse sum indeg (<em>ISI</em>) index of the graph <em>G</em> is defined as <em>ISI</em>(<em>G</em>)=∑_{<em>i</em>∼<em>j</em>}(<em>d_id_j</em>)/(<em>d_i</em>+<em>d_j</em>). Some new upper bounds for the <em>ISI</em> index are obtained in this paper.

In 1960, Sierpiński proved that there exist infinitely many odd positive integers k such that k · 2n + 1 is composite for all integers n ≥ 0. Variations of this problem, using polynomials with integer coefficients, and considering reducibility over the rationals, have been investigated by several authors. In particular, if S is the set of all positive integers d for which there exists a polynomial f(x) ∈ ℤ[x], with f(1) ≠ -d, such that f(x)xn + d is reducible over the rationals for all integers n ≥ 0, then what are the elements of S? Interest in this problem stems partially from the fact that if S contains an odd integer, then a question of Erdös and Selfridge concerning the existence of an odd covering of the integers would be resolved. Filaseta has shown that S contains all positive integers d ≡ 0 (mod 4), and until now, nothing else was known about the elements of S. In this paper, we show that S contains infinitely many positive integers d ≡ 6 (mod 12). We also consider the corresponding problem over 𝔽p, and in that situation, we show 1 ∈ S for all primes p.

Asymptotic enumeration of 0–1 matrices with equal row sums and equal column sums

Asymptotic behavior in time of the solutions of three nonlinear partial differential equations

This paper is an extension of a recent work done by the author [4] and here the sums of alternating series of odd powers (up to fifteen) of the reciprocals of odd positive integers are computed. Following this method, the sum of the series of any higher power could be calculated. In the process of computing these sums, the sums of the series of even powers of reciprocals of odd positive integers have been reestablished and enabled the author to compute the values of Riemann’s zeta function for even positive integers.

Noether's problem for some semidirect products

A graph is a mathematical structure consisting of a set of objects, which we call vertices, and links between pairs of objects, which we call edges. Graphs are used to model many problems arising in areas such as physics, sociology, and computer science. It is partially because of the simplicity of the definition of a graph that the concept can be so widely used. Nevertheless, when applied to a particular task, it is not always necessary to study graphs in all their generality, and it can be convenient to studying them from a restricted point of view. Restriction can come from requiring graphs to be embeddable in a particular surface, to admit certain types of decompositions, or by forbidding some substructure. A collection of graphs satisfying a fixed restriction forms a class of graphs. Many important classes of graphs satisfy that graphs belonging to it cannot have many edges in comparison with the number of vertices. Such is the case of classes with an upper bound on the maximum degree, and of classes excluding a fixed minor. Recently, the notion of classes with bounded expansion was introduced by Neˇsetˇril and Ossona de Mendez [62], as a generalisation of many important types of sparse classes. In this thesis we study chromatic and structural properties of classes with bounded expansion. We say a graph is k-degenerate if each of its subgraphs has a vertex of degree at most k. The degeneracy is thus a measure of the density of a graph. This notion has been generalised with the introduction, by Kierstead and Yang [47], of the generalised colouring numbers. These parameters have found applications in many areas of Graph Theory, including a characterisation of classes with bounded expansion. One of the main results of this thesis is a series of upper bounds on the generalised colouring numbers, for different sparse classes of graphs, such as classes excluding a fixed complete minor, classes with bounded genus and classes with bounded tree-width. We also study the following problem: for a fixed positive integer p, how many colours do we need to colour a given graph in such a way that vertices at distance exactly p get different colours? When considering classes with bounded expansion, we improve dramatically on the previously known upper bounds for the number of colours needed. Finally, we introduce a notion of addition of graph classes, and show various cases in which sparse classes can be summed so as to obtain another sparse class.

For a given positive integer t we consider graphs having maximal independent sets of precisely t distinct cardinalities and restrict our attention to those that have no vertices of degree one. In the situation when t is four or larger and the length of the shortest cycle is at least 6t − 6, we completely characterize such graphs.

Given a set $\delta = \{ d_0 ,d_1 , \cdots ,d_{n - 1} \} $ of positive integers and a positive integer p, the necessary and sufficient conditions are presented for existence of a p-connected n-vertex graph (which allows parallel edges but no self-loops) whose vertex degrees are $d_0 ,d_1 , \cdots ,d_{n - 1} $. Some related problems are discussed.

The Mathieu functions of integral order [1] are the solutions with period π or 2π of the equationThe eigenvalues associated with the functions ceN and seN, where N is a positive integer, denoted by aN and bN respectively, reduce toaN = bN = N2when q is zero. The quantities aN and bN can be expanded in powers of q, but the explicit construction of high order coefficients is very tedious. In some applications the quantity of most interest is aN – bN, which may be called the “width of the unstable zone“. It is the object of this note to derive a general formula for the leading term in the expansion of this quantity, namelySuppose first that N is an odd integer. Then there is an expansionwhereThese functions π satisfyandOn Substituting (3) in (1), one obtains the algebraic equationwhereExplicitly,{11} = q{lm} = 0 otherwise.

Let a be a positive integer with $a>1$ , and let $(x, y, n)$ be a positive integer solution of the equation $x^{2}+a^{2}=y^{n}$ , $\gcd(x, y)=1$ , $n>2$ . Using Baker’s method, we prove that, for any positive number ϵ, if n is an odd integer with $n>C(\epsilon)$ , where $C(\epsilon)$ is an effectively computable constant depending only on ϵ, then $n<(2+\epsilon)(\log a)/\log y$ . Owing to the obvious fact that every solution $(x, y, n)$ of the equation satisfies $n>2(\log a)/\log y$ , the above upper bound is optimal.

In the present paper I obtain an extension of the so-called Eisenstein theorem which is proved by means of RiemannStieltjes integration. THEOREM. Let n be a positive integer >2. Let P1.P2, p.. ,Pn be odd integers and relatively prime in pairs. Then n (pi-1)/2 n P 1 n i=1 x=1 k=l;k*i L Pi k=1 PROOF. The method of proof is Riemann-Stieltjes integration. Consider the integrals r1/2 1 = [P1X] [p2x] ... [Pn-X] d[pnx], 1/2 2 = [PnX] d[p1x] [P2x] . [Pn-lX] First we observe that the greatest integer functions [plx], [p2x],*** [pAx] have no discontinuities in common on the interval O<x l in view of the condition on the integers Pl, P2, , Pn. Second, the discontinuities of [p x], i=1, 2, , n, on the interval O<x_ are at x=klpi, k=1, 2, * , *21(pi-1), and the jump of [pix] at each of these discontinuities is equal to 1. Hence the value of Y1 is 2 [plx] [p2x] * [pn_x], k = 1, 2, . . . 21(pn1) x=k1p, or (Pn-l)/2 rPlXrP2Xl -Pn 1X Consider the second integral. The discontinuities of [p1x] [p2x] . . [pn_IX] are at x=klpi, k=1,2, ,21(pi-l), i=1,2, ,n-1. Since there are no common discontinuities and the jumps of [pix] at the discontinuities k/pi are equal to 1, the jump of [plx] [p2x] ... [Pn-IX] at each Received by the editors May 9, 1973. AMS (MOS) subject classifications (1970). Primary 1OA20, 1OA25. ? American Mathematical Society 1974 69 This content downloaded from 157.55.39.243 on Wed, 05 Oct 2016 04:52:27 UTC All use subject to http://about.jstor.org/terms

Erdős and Odlyzko proved that odd integers k such that k2n+1 is prime for some positive integer n have a positive lower density. In this paper, we characterize all arithmetic progressions in which natural numbers that can be expressed in the form (p−1)2−n (where p is a prime number) have a positive proportion. We also prove that an arithmetic progression consisting of odd numbers can be obtained from a covering system if and only if those integers in such a progression which can be expressed in the form (p−1)2−n have an asymptotic density of zero.

Let b and c be fixed coprime odd positive integers with min{b, c} > 1. In this paper, a classification of all positive integer solutions (x, y, z) of the equation 2x + by = cz is given. Further, by an elementary approach, we prove that if c = b + 2, then the equation has only the positive integer solution (x, y, z) = (1,1, 1), except for (b, x, y, z) = (89,13,1, 2) and (2r − 1, r + 2,2, 2), where r is a positive integer with r ≥ 2.