Modular Irregular Labeling on Double-Star and Friendship Graphs

A modular irregular graph is a graph that admits a modular irregular labeling. A modular irregular labeling of a graph G of order n is a mapping of the set of edges of the graph to 1,2 , … , k such that the weights of all vertices are different. The vertex weight is the sum of its incident edge labels, and all vertex weights are calculated with the sum modulo n . The modular irregularity strength is the minimum largest edge label such that a modular irregular labeling can be done. In this paper, we construct a modular irregular labeling of two classes of graphs that are biregular; in this case, the regular double-star graph and friendship graph classes are chosen. Since the modular irregularity strength of the friendship graph also holds the minimal irregularity strength, then the labeling is also an irregular labeling with the same strength as the modular case.

The diachromatic number of double star graph

We are interested in the extension for the concept of complete colouring for oriented graph G → that has been proposed in many different notions by several authors (Edwards, Sopena, and Araujo-Pardo in 2013, 2014, and 2018, respectively). An oriented colouring is complete if for every ordered pair of colours, at least one arc in G → whose endpoints are coloured with these colours. The diachromatic number, dac ( G → ) , is the greatest number of colours in a complete oriented colouring. In this paper, we establish the formula of diachromatic numbers for double star graph, k 1 , n , n → , over all possible orientations on the graph. In particular, if din (u) = 0 (resp. dout(u) = 0)and din (wi ) = 1 (resp. dout (w 1) = 1) for all i, then dac ( k 1 , n , n → ) = ⌊ n ⌋ + 1 , where u is the internal vertex and w i , i ∈ {1,…, n}, is the pendant vertices of the digraph.

On the Eventual Exponential Positivity of Some Tree Sign Patterns

An n×n matrix A is called eventually exponentially positive (EEP) if etA=∑k=0∞tkAkk!>0 for all t≥t0, where t0≥0. A matrix whose entries belong to the set {+,−,0} is called a sign pattern. An n×n sign pattern A is called potentially eventually exponentially positive (PEEP) if there exists some real matrix realization A of A that is EEP. Characterizing the PEEP sign patterns is a longstanding open problem. In this article, A is called minimally potentially eventually exponentially positive (MPEEP), if A is PEEP and no proper subpattern of A is PEEP. Some preliminary results about MPEEP sign patterns and PEEP sign patterns are established. All MPEEP sign patterns of orders n≤3 are identified. For the n×n tridiagonal sign patterns Tn, we show that there exists exactly one MPEEP tridiagonal sign pattern Tno. Consequently, we classify all PEEP tridiagonal sign patterns as the superpatterns of Tno. We also classify all PEEP star sign patterns Sn and double star sign patterns DS(n,m) by identifying all the MPEEP star sign patterns and the MPEEP double star sign patterns, respectively.

Theory of chaos synchronization and quasi-period synchronization of an all optic 2n-D LAN

Theory of chaos synchronization and quasi-period synchronization of an all optics local area network (O-LAN) is deeply studied and discussed, where two coupled-lasers are used as network’s double-star and the other single-lasers are used as network nodes. The LAN operates double-star lasers to drive node lasers in two links to perform a 2n−D (n is a positive integer, dimensions (D)) laser network. The O-LAN has the characteristics of an all optics LAN with double-center and two link nodes. Our theoretical and numerical results prove that the double-center lasers can obtain their synchronizations with each laser in two link nodes. A route to chaos after a quasi-period bifurcation is analyzed to illustrate dynamics distribution region of O-LAN. We find five quasi-period regions, four chaos regions, where there is a region where instability mixes with the first chaos, and a stable region. We find also that O-LAN can obtain its parallel multi-dynamics synchronizations, such as cycle-one synchronization, cycle-2 synchronization, cycle-3 synchronization, cycle-4 synchronization, cycle-5 synchronization, other quasi-period synchronization and chaos synchronization, shown in two links of O-LAN by shifting the currents of the lasers in one link. The theory of all optics LAN and its obtained results are useful to study on complex dynamic system, optics network, artificial intelligence, chaos and its synchronization.

Extending of Edge Even Graceful Labeling of Graphs to Strong r -Edge Even Graceful Labeling

Edge even graceful labeling of a graph G with p vertices and q edges is a bijective f from the set of edge E G to the set of positive integers 2,4 , … , 2 q such that all the vertex labels f ∗ V G , given by f ∗ u = ∑ u v ∈ E G f u v mod 2 k , where k = max p , q , are pairwise distinct. There are many graphs that do not have edge even graceful labeling, so in this paper, we have extended the definition of edge even graceful labeling to r -edge even graceful labeling and strong r -edge even graceful labeling. We have obtained the necessary conditions for more path-related graphs and cycle-related graphs to be an r -edge even graceful graph. Furthermore, the minimum number r for which the graphs: tortoise graph, double star graph, ladder and diagonal ladder graphs, helm graph, crown graph, sunflower graph, and sunflower planar graph, have an r -edge even graceful labeling was found. Finally, we proved that the even cycle C 2 n has a strong 2 -edge even graceful labeling when n is even.

Measure by Measure, they touched the heaven

The measure of distances is a recurring theme in astrophysics. The interpretation of the light coming from a luminous object in the sky can be very different depending on the distance of the object. Two stars or galaxies may each have a different real brightness, although they may look similar. The correct measures were determined by women computers a century ago. Special mention is due to Williamina Fleming, who supervised an observatory for 30 years and worked on the first system to classify stars by spectrum. Antonia Maury helped locate the first double star and developed a new star classification system. Henrietta Leavitt determined a law to calculate stellar distances. The most famous of the Harvard computers was Annie Jump Cannon. An expert in photography, she catalogued over 350,000 stars and expanded the classification system used today, but it was Henrietta Leavitt who left an indelible mark by discovering a law for the determination of stellar distances. In the same period, Italian women computers began to collaborate in observatories, but their tracks are obfuscated.

Restrained Italian domination in graphs

For a graph G = (V(G), E(G)), an Italian dominating function (ID function) f : V(G) → {0,1,2} has the property that for every vertex v ∈ V(G) with f(v) = 0, either v is adjacent to a vertex assigned 2 under f or v is adjacent to least two vertices assigned 1 under f. The weight of an ID function is ∑v∈V(G) f(v). The Italian domination number is the minimum weight taken over all ID functions of G. In this paper, we initiate the study of a variant of ID functions. A restrained Italian dominating function (RID function) f of G is an ID function of G for which the subgraph induced by {v ∈ V(G) | f(v) = 0} has no isolated vertices, and the restrained Italian domination number γrI (G) is the minimum weight taken over all RID functions of G. We first prove that the problem of computing this parameter is NP-hard, even when restricted to bipartite graphs and chordal graphs as well as planar graphs with maximum degree five. We prove that γrI(T) for a tree T of order n ≥ 3 different from the double star S2,2 can be bounded from below by (n + 3)/2. Moreover, all extremal trees for this lower bound are characterized in this paper. We also give some sharp bounds on this parameter for general graphs and give the characterizations of graphs G with small or large γrI (G).