2,922 research outputs found

    On Total Irregularity Strength of Double-Star and Related Graphs

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    AbstractLet G = (V, E) be a simple and undirected graph with a vertex set V and an edge set E. A totally irregular total k-labeling f : V ∪ E → {1, 2,. . ., k} is a labeling of vertices and edges of G in such a way that for any two different vertices x and x1, their weights and are distinct, and for any two different edges xy and x1y1 their weights f (x) + f (xy) + f (y) and f (x1) + f (x1y1) + f (y1) are also distinct. A total irregularity strength of graph G, denoted byts(G), is defined as the minimum k for which G has a totally irregular total k-labeling. In this paper, we determine the exact value of the total irregularity strength for double-star S n,m, n, m ≥ 3 and graph related to it, that is a caterpillar S n,2,n, n ≥ 3. The results are and ts(S n,2,n) = n

    Total absolute difference edge irregularity strength of some families of graphs

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    A total labeling ξ is defined to be an edge irregular total absolute difference k-labeling of the graph G if for every two different edges e and f of G there is wt(e) ≠ wt(f) where weight of an edge e = xy is defined as wt(e) = |ξ(e) − ξ(x) − ξ(y)|. The minimum k for which the graph G has an edge irregular total absolute difference labeling is called the total absolute difference edge irregularity strength of the graph G, tades(G). In this paper, we determine the total absolute difference edge irregularity strength of the precise values for some families of graphs.Publisher's Versio

    Further Results on (a, d) -total Edge Irregularity Strength of Graphs

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    ليكن  رسمًا بيانيًا بسيطًا على رؤوس l وحواف m مع إجمالي h -  وضع العلامات  . فان   تسمى (ا,د)- وسم غير منتظم للحافة الإجمالية إذا وجد تطابق متقابل وليكن   معرفة بواسطة   لكل   , حيث  . كذلك قيمة  يقال لها وزن الحافة . يشار الى (ا,د)-اجمالي قوة عدم انتظام الحواف للرسم البياني G ب  وهي اقل h التي يقبلها G   للحافة -(ا,د) الغير منتظمة للعلامة-h . في هذه المقالة تم فحص,  لبعض عائلات الرسم البياني الشائعة. بالاضافة الى ذلك تم حل المسالة المفتوحة  بشكل ايجابي. م تسمى ρ (أ ، د) - وسم غير منتظم للحافة الإجمالية إذا كان هناك تطابق واحد لواحد ، قل ψ: E (G) → {a ، a + d ، a + 2d ،… + a + (m- 1) د} محدد بواسطة ψ (uv) = ρ (u) + ρ (v) + ρ (uv) لجميع uv∈E (G) ، حيث a≥3 ، d≥2. أيضًا ، يُقال إن القيمة ψ (uv) هي وزن حافة الأشعة فوق البنفسجية. يشار إلى قوة عدم انتظام الحافة الإجمالية (أ ، د) للرسم البياني G بواسطة (a ، d) -tes (G) وهي أقل h التي يقبلها G (أ ، د) - علامة h غير منتظمة للحافة. في هذه المقالة ، يتم فحص (أ ، د) -tes (G) لبعض عائلات الرسم البياني الشائعة. بالإضافة إلى ذلك ، يتم حل المشكلة المفتوحة (3،2) - tes (K_ (m ، n)) ، m ، n> 2 بشكل إيجابي.Consider a simple graph   on vertices and edges together with a total  labeling . Then ρ is called total edge irregular labeling if there exists a one-to-one correspondence, say  defined by  for all  where  Also, the value  is said to be the edge weight of . The total edge irregularity strength of the graph G is indicated by  and is the least  for which G admits   edge irregular h-labeling.  In this article,   for some common graph families are examined. In addition, an open problem is solved affirmatively

    SOME CARTESIAN PRODUCTS OF A PATH AND PRISM RELATED GRAPHS THAT ARE EDGE ODD GRACEFUL

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    Let GG be a connected undirected simple graph of size qq and let kk be the maximum number of its order and its size. Let ff be a bijective edge labeling which codomain is the set of odd integers from 1 up to 2q12q-1. Then ff is called an edge odd graceful on GG if the weights of all vertices are distinct, where the weight of a vertex vv is defined as the sum mod(2k)mod(2k) of all labels of edges incident to vv. Any graph that admits an edge odd graceful labeling is called an edge odd graceful graph. In this paper, some new graph classes that are edge odd graceful are presented, namely some cartesian products of path of length two and some circular related graphs

    Connectivity Influences on Nonlinear Dynamics in Weakly-Synchronized Networks: Insights from Rössler Systems, Electronic Chaotic Oscillators, Model and Biological Neurons

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    Natural and engineered networks, such as interconnected neurons, ecological and social networks, coupled oscillators, wireless terminals and power loads, are characterized by an appreciable heterogeneity in the local connectivity around each node. For instance, in both elementary structures such as stars and complex graphs having scale-free topology, a minority of elements are linked to the rest of the network disproportionately strongly. While the effect of the arrangement of structural connections on the emergent synchronization pattern has been studied extensively, considerably less is known about its influence on the temporal dynamics unfolding within each node. Here, we present a comprehensive investigation across diverse simulated and experimental systems, encompassing star and complex networks of Rössler systems, coupled hysteresis-based electronic oscillators, microcircuits of leaky integrate-and-fire model neurons, and finally recordings from in-vitro cultures of spontaneously-growing neuronal networks. We systematically consider a range of dynamical measures, including the correlation dimension, nonlinear prediction error, permutation entropy, and other information-theoretical indices. The empirical evidence gathered reveals that under situations of weak synchronization, wherein rather than a collective behavior one observes significantly differentiated dynamics, denser connectivity tends to locally promote the emergence of stronger signatures of nonlinear dynamics. In deterministic systems, transition to chaos and generation of higher-dimensional signals were observed; however, when the coupling is stronger, this relationship may be lost or even inverted. In systems with a strong stochastic component, the generation of more temporally-organized activity could be induced. These observations have many potential implications across diverse fields of basic and applied science, for example, in the design of distributed sensing systems based on wireless coupled oscillators, in network identification and control, as well as in the interpretation of neuroscientific and other dynamical data

    Advances in Discrete Applied Mathematics and Graph Theory

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    The present reprint contains twelve papers published in the Special Issue “Advances in Discrete Applied Mathematics and Graph Theory, 2021” of the MDPI Mathematics journal, which cover a wide range of topics connected to the theory and applications of Graph Theory and Discrete Applied Mathematics. The focus of the majority of papers is on recent advances in graph theory and applications in chemical graph theory. In particular, the topics studied include bipartite and multipartite Ramsey numbers, graph coloring and chromatic numbers, several varieties of domination (Double Roman, Quasi-Total Roman, Total 3-Roman) and two graph indices of interest in chemical graph theory (Sombor index, generalized ABC index), as well as hyperspaces of graphs and local inclusive distance vertex irregular graphs

    The Eccentric-distance Sum of Some Graphs

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    Let G=(V,E)G = (V,E) be a simple connected graph. Theeccentric-distance sum of GG is defined as\xi^{ds}(G) =\ds\sum_{\{u,v\}\subseteq V(G)} [e(u)+e(v)] d(u,v), where e(u)e(u) %\dsis the eccentricity of the vertex uu in GG and d(u,v)d(u,v) is thedistance between uu and vv. In this paper, we establish formulaeto calculate the eccentric-distance sum for some graphs, namelywheel, star, broom, lollipop, double star, friendship, multi-stargraph and the join of Pn2P_{n-2} and P2P_2

    Normalised Degree Variance

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    Finding graph indices which are unbiased to network size and density is of high importance both within a given field and across fields for enhancing comparability of modern network science studies. The degree variance is an important metric for characterising network degree heterogeneity. Here, we provide an analytically valid normalisation of degree variance to replace previous normalisations which are either invalid or not applicable to all networks. It is shown that this normalisation provides equal values for graphs and their complements; it is maximal in the star graph (and its complement); and its expected value is constant with respect to density for Erd\"os-R\'enyi (ER) random graphs of the same size. We strengthen these results with model observations in ER random graphs, random geometric graphs, scale-free networks, random hierarchy networks and resting-state brain networks, showing that the proposed normalisation is generally less affected by both network size and density than previous normalisation attempts. The closed form expression proposed also benefits from high computational efficiency and straightforward mathematical analysis. Analysis of 184 real-world binary networks across different disciplines shows that normalised degree variance is not correlated with average degree and is robust to node and edge subsampling. Comparisons across subdomains of biological networks reveals greater degree heterogeneity among brain connectomes and food webs than in protein interaction networks.Comment: 16 pages, 6 figure
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