scholarly journals The signless Laplacian coefficients and the incidence energy of graphs with a given bipartition

Filomat ◽  
2020 ◽  
Vol 34 (12) ◽  
pp. 4215-4232
Author(s):  
Lei Zhong ◽  
Wen-Huan Wang
Keyword(s):  
Minimal Element ◽  
Unicyclic Graph ◽  
The Other ◽  
Unicyclic Graphs ◽  
Graph Transformations ◽  
Signless Laplacian ◽  
Positive Integers ◽  
Quasi Ordering

We consider two classes of the graphs with a given bipartition. One is trees and the other is unicyclic graphs. The signless Laplacian coefficients and the incidence energy are investigated for the sets of trees/unicyclic graphs with n vertices in which each tree/unicyclic graph has an (n1,n2)-bipartition, where n1 and n2 are positive integers not less than 2 and n1+n2 = n. Four new graph transformations are proposed for studying the signless Laplacian coefficients. Among the sets of trees/unicyclic graphs considered, we obtain exactly, for each, the minimal element with respect to the quasi-ordering according to their signless Laplacian coefficients and the element with the minimal incidence energies.

scholarly journals The sum of the first two largest signless Laplacian eigenvalues of trees and unicyclic graphs

2019 ◽  
Vol 35 ◽  
pp. 449-467
Author(s):  
Zhibin Du
Keyword(s):  
Unicyclic Graph ◽  
Unicyclic Graphs ◽  
Laplacian Eigenvalues ◽  
Signless Laplacian ◽  
Pendent Vertices ◽  
Eigenvalues Of Graphs

Let $G$ be a graph on $n$ vertices with $e(G)$ edges. The sum of eigenvalues of graphs has been receiving a lot of attention these years. Let $S_2 (G)$ be the sum of the first two largest signless Laplacian eigenvalues of $G$, and define $f(G) = e (G) +3 - S_2 (G)$. Oliveira et al. (2015) conjectured that $f(G) \geqslant f(U_{n})$ with equality if and only if $G \cong U_n$, where $U_n$ is the $n$-vertex unicyclic graph obtained by attaching $n-3$ pendent vertices to a vertex of a triangle. In this paper, it is proved that $S_2(G) < e(G) + 3 -\frac{2}{n}$ when $G$ is a tree, or a unicyclic graph whose unique cycle is not a triangle. As a consequence, it is deduced that the conjecture proposed by Oliveira et al. is true for trees and unicyclic graphs whose unique cycle is not a triangle.

scholarly journals Wiener Complexity versus the Eccentric Complexity

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 79
Author(s):  
Martin Knor ◽  
Riste Škrekovski
Keyword(s):  
Unicyclic Graph ◽  
The Other ◽  
Unicyclic Graphs ◽  
Cyclomatic Number ◽  
Open Question ◽  
Sum Of Distances

Let wG(u) be the sum of distances from u to all the other vertices of G. The Wiener complexity, CW(G), is the number of different values of wG(u) in G, and the eccentric complexity, Cec(G), is the number of different eccentricities in G. In this paper, we prove that for every integer c there are infinitely many graphs G such that CW(G)−Cec(G)=c. Moreover, we prove this statement using graphs with the smallest possible cyclomatic number. That is, if c≥0 we prove this statement using trees, and if c<0 we prove it using unicyclic graphs. Further, we prove that Cec(G)≤2CW(G)−1 if G is a unicyclic graph. In our proofs we use that the function wG(u) is convex on paths consisting of bridges. This property also promptly implies the already known bound for trees Cec(G)≤CW(G). Finally, we answer in positive an open question by finding infinitely many graphs G with diameter 3 such that Cec(G)<CW(G).

Pengaruh Unsur Budaya Lokal dalam Ungkapan Berbahasa Perancis

Metahumaniora ◽  
2017 ◽  
Vol 7 (3) ◽  
pp. 378
Author(s):  
Vincentia Tri Handayani
Keyword(s):  
Mental Representation ◽  
Cultural Context ◽  
Oral Tradition ◽  
Minimal Element ◽  
The Other ◽  
Local Culture ◽  
Community Groups ◽  
Cultural Community ◽  
The Difference ◽  
Complex Words

AbstrakFolklor yang menghasilkan tradisi lisan merupakan perwujudan budaya yang lahirdari pengalaman kelompok masyarakat. Salah satu bentuk tradisi lisan adalah ungkapan yangmengandung unsur budaya lokal dalam konstruksinya yang tidak dimiliki budaya lainnya.Ungkapan idiomatis memberikan warna pada bahasa melalui penggambaran mental. Dalambahasa Perancis, ungkapan dapat berupa locution dan expression. Perbedaan motif acuansuatu ungkapan dapat terlihat dari pengaruh budaya masyarakat pengguna bahasa. Sebuahleksem tidak selalu didefinisikan melalui unsur minimal, tidak juga melalui kata-kata,baik kata dasar atau kata kompleks, namun dapat melalui kata-kata beku yang maknanyatetap. Hubungan analogis dari makna tambahan yang ada pada suatu leksem muncul dariidentifikasi semem yang sama. Semem tersebut mengarah pada term yang diasosiasikan danyang diperkaya melalui konteks (dalam ungkapan berhubungan dengan konteks budaya).Kata kunci: folklor, ungkapan, struktur, makna idiomatis, kebudayaanAbstractFolklore which produces the oral tradition is a cultural manifestation born out theexperience of community groups. One form of the oral tradition is a phrase that containsthe elements of local culture in its construction that is not owned the other culture. Theidiomatic phrase gives the color to the language through the mental representation. InFrench, the expression can consist of locution and expression. The difference motivesreference of an expression can be seen from the influence of the cultural community thelanguage users. A lexeme is not always defined through a minimal element, nor throughwords, either basic or complex words, but can be through the frost words whose meaningsare fixed. The analogical connection of the additional meanings is on a lexeme arises fromthe identification of the same meaning. The meaning ‘semem’ leads to the associated termsand which are enriched through the context (in idiom related to the cultural context).Keywords : folklore, idioms, structure, idiom meaning, cultureI PENDAHULUAN

scholarly journals On the Turán Properties of Infinite Graphs

10.37236/771 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Andrzej Dudek ◽  
Vojtěch Rödl
Keyword(s):  
Infinite Graph ◽  
Infinite Graphs ◽  
The Other ◽  
Other Hand ◽  
Turan's Theorem ◽  
Vertex Set ◽  
Turán’S Theorem ◽  
Positive Integers

Let $G^{(\infty)}$ be an infinite graph with the vertex set corresponding to the set of positive integers ${\Bbb N}$. Denote by $G^{(l)}$ a subgraph of $G^{(\infty)}$ which is spanned by the vertices $\{1,\dots,l\}$. As a possible extension of Turán's theorem to infinite graphs, in this paper we will examine how large $\liminf_{l\rightarrow \infty} {|E(G^{(l)})|\over l^2}$ can be for an infinite graph $G^{(\infty)}$, which does not contain an increasing path $I_k$ with $k+1$ vertices. We will show that for sufficiently large $k$ there are $I_k$–free infinite graphs with ${1\over 4}+{1\over 200} < \liminf_{l\rightarrow \infty} {|E(G^{(l)})|\over l^2}$. This disproves a conjecture of J. Czipszer, P. Erdős and A. Hajnal. On the other hand, we will show that $\liminf_{l\rightarrow \infty} {|E(G^{(l)})|\over l^2}\le{1\over 3}$ for any $k$ and such $G^{(\infty)}$.

scholarly journals Equations with powers of singular moduli

2019 ◽  
Vol 15 (03) ◽  
pp. 445-468 ◽  
Author(s):  
Antonin Riffaut
Keyword(s):  
Rational Number ◽  
The Other ◽  
Other Hand ◽  
Singular Moduli ◽  
Cm Points ◽  
The One ◽  
Straight Lines ◽  
Positive Integers

We treat two different equations involving powers of singular moduli. On the one hand, we show that, with two possible (explicitly specified) exceptions, two distinct singular moduli [Formula: see text] such that the numbers [Formula: see text], [Formula: see text] and [Formula: see text] are linearly dependent over [Formula: see text] for some positive integers [Formula: see text], must be of degree at most [Formula: see text]. This partially generalizes a result of Allombert, Bilu and Pizarro-Madariaga, who studied CM-points belonging to straight lines in [Formula: see text] defined over [Formula: see text]. On the other hand, we show that, with obvious exceptions, the product of any two powers of singular moduli cannot be a non-zero rational number. This generalizes a result of Bilu, Luca and Pizarro-Madariaga, who studied CM-points belonging to a hyperbola [Formula: see text], where [Formula: see text].

Distance spectral radius of unicyclic graphs with fixed maximum degree

2019 ◽  
Vol 19 (04) ◽  
pp. 2050068
Author(s):  
Hezan Huang ◽  
Bo Zhou
Keyword(s):  
Spectral Radius ◽  
Connected Graph ◽  
Maximum Degree ◽  
Distance Matrix ◽  
The Other ◽  
Connected Graphs ◽  
Largest Eigenvalue ◽  
Unicyclic Graphs ◽  
The Largest Eigenvalue

The distance spectral radius of a connected graph is the largest eigenvalue of its distance matrix. For integers [Formula: see text] and [Formula: see text] with [Formula: see text], we prove that among the connected graphs on [Formula: see text] vertices of given maximum degree [Formula: see text] with at least one cycle, the graph [Formula: see text] uniquely maximizes the distance spectral radius, where [Formula: see text] is the graph obtained from the disjoint star on [Formula: see text] vertices and path on [Formula: see text] vertices by adding two edges, one connecting the star center with a path end, and the other being a chord of the star.

scholarly journals ON ISOTOPISMS OF COMMUTATIVE PRESEMIFIELDS AND CCZ-EQUIVALENCE OF FUNCTIONS

2011 ◽  
Vol 22 (06) ◽  
pp. 1243-1258 ◽  
Author(s):  
LILYA BUDAGHYAN ◽  
TOR HELLESETH
Keyword(s):  
Equivalence Relation ◽  
The Other ◽  
Ccz Equivalence ◽  
Positive Divisor ◽  
Positive Integers

A function F from Fpnto itself is planar if for any [Formula: see text] the function F(x+a)-F(x) is a permutation. CCZ-equivalence is the most general known equivalence relation of functions preserving planar property. This paper considers two possible extensions of CCZ-equivalence for functions over fields of odd characteristics, one proposed by Coulter and Henderson and the other by Budaghyan and Carlet. We show that the second one in fact coincides with CCZ-equivalence, while using the first one we generalize one of the known families of PN functions. In particular, we prove that, for any odd prime p and any positive integers n and m, the indicators of the graphs of functions F and F' from Fpnto Fpmare CCZ-equivalent if and only if F and F′ are CCZ-equivalent.We also prove that, for any odd prime p, CCZ-equivalence of functions from Fpnto Fpm, is strictly more general than EA-equivalence when n ≥ 3 and m is greater or equal to the smallest positive divisor of n different from 1.

scholarly journals On the number of arithmetic formulas

2015 ◽  
Vol 11 (04) ◽  
pp. 1099-1106 ◽  
Author(s):  
Carlo Sanna
Keyword(s):  
The Other ◽  
Repeated Application ◽  
Positive Integers ◽  
Arithmetic Expressions

For each positive integers n, let g(n) be the number of arithmetic expressions evaluating to n and involving only the constant 1, additions and multiplications, with the restriction that multiplication by 1 is not allowed. We consider two arithmetic expressions to be equal if one can be obtained from the other through a repeated application of the commutative and associative properties. We give an algorithm to compute g(n) and prove that [Formula: see text], as n → +∞, where β ≔ log(24)/24.

Arithmetic functions monotonic at consecutive arguments

2014 ◽  
Vol 51 (2) ◽  
pp. 155-164
Author(s):  
Jean-Marie Koninck ◽  
Florian Luca
Keyword(s):  
Large Class ◽  
The Other ◽  
Arithmetic Functions ◽  
Arbitrary Integer ◽  
Other Hand ◽  
Positive Integers

For a large class of arithmetic functions f, it is possible to show that, given an arbitrary integer κ ≤ 2, the string of inequalities f(n + 1) < f(n + 2) < … < f(n + κ) holds for in-finitely many positive integers n. For other arithmetic functions f, such a property fails to hold even for κ = 3. We examine arithmetic functions from both classes. In particular, we show that there are only finitely many values of n satisfying σ2(n − 1) < σ2 < σ2(n + 1), where σ2(n) = ∑d|nd2. On the other hand, we prove that for the function f(n) := ∑p|np2, we do have f(n − 1) < f(n) < f(n + 1) in finitely often.

scholarly journals A further correspondence between $(bc,\bar{b})$-parking functions and $(bc,\bar{b})$-forests

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Heesung Shin ◽  
Jiang Zeng
Keyword(s):  
The Other ◽  
Parking Functions ◽  
Fixed Sequence ◽  
Parking Function ◽  
International Audience ◽  
Positive Integers ◽  
Appl Math

International audience For a fixed sequence of $n$ positive integers $(a,\bar{b}) := (a, b, b,\ldots, b)$, an $(a,\bar{b})$-parking function of length $n$ is a sequence $(p_1, p_2, \ldots, p_n)$ of positive integers whose nondecreasing rearrangement $q_1 \leq q_2 \leq \cdots \leq q_n$ satisfies $q_i \leq a+(i-1)b$ for any $i=1,\ldots, n$. A $(a,\bar{b})$-forest on $n$-set is a rooted vertex-colored forests on $n$-set whose roots are colored with the colors $0, 1, \ldots, a-1$ and the other vertices are colored with the colors $0, 1, \ldots, b-1$. In this paper, we construct a bijection between $(bc,\bar{b})$-parking functions of length $n$ and $(bc,\bar{b})$-forests on $n$-set with some interesting properties. As applications, we obtain a generalization of Gessel and Seo's result about $(c,\bar{1})$-parking functions [Ira M. Gessel and Seunghyun Seo, Electron. J. Combin. $\textbf{11}$(2)R27, 2004] and a refinement of Yan's identity [Catherine H. Yan, Adv. Appl. Math. $\textbf{27}$(2―3):641―670, 2001] between an inversion enumerator for $(bc,\bar{b})$-forests and a complement enumerator for $(bc,\bar{b})$-parking functions. Soit $(a,\bar{b}) := (a, b, b,\ldots, b)$ une suite d'entiers positifs. Une $(a,\bar{b})$-fonction de parking est une suite $(p_1, p_2, \ldots, p_n)$ d'entiers positives telle que son réarrangement non décroissant $q_1 \leq q_2 \leq \cdots \leq q_n$ satisfait $q_i \leq a+(i-1)b$ pour tout $i=1,\ldots, n$. Une $(a,\bar{b})$-forêt enracinée sur un $n$-ensemble est une forêt enracinée dont les racines sont colorées avec les couleurs $0, 1, \ldots, a-1$ et les autres sommets sont colorés avec les couleurs $0, 1, \ldots, b-1$. Dans cet article, on construit une bijection entre $(bc,\bar{b})$-fonctions de parking et $(bc,\bar{b})$-forêts avec des des propriétés intéressantes. Comme applications, on obtient une généralisation d'un résultat de Gessel-Seo sur $(c,\bar{1})$-fonctions de parking [Ira M. Gessel and Seunghyun Seo, Electron. J. Combin. $\textbf{11}$(2)R27, 2004] et une extension de l'identité de Yan [Catherine H. Yan, Adv. Appl. Math. $\textbf{27}$(2―3):641―670, 2001] entre l'énumérateur d'inversion de $(bc,\bar{b})$-forêts et l'énumérateur complémentaire de $(bc,\bar{b})$-fonctions de parking.

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