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Max-plus algebra is the similarity of the classical linear algebra with two binary operations, maximum and addition. The notation Ax = Bx, where A, B are given (interval) matrices, represents (interval) two-sided (max, plus)-linear system. For the solvability of Ax = Bx, there are some pseudopolynomial algorithms, but a polynomial algorithm is still waiting for an appearance. The paper deals with the analysis of solvability of two-sided (max, plus)-linear equations with inexact (interval) data. The purpose of the paper is to get efficient necessary and sufficient conditions for solvability of the interval systems using the property of the solution set of the non-interval system Ax = Bx. The main contribution of the paper is a transformation of weak versions of solvability to either subeigenvector problems or to non-interval two-sided (max, plus)-linear systems and obtaining the equivalent polynomially checked conditions for the strong versions of solvability.

The Ewens–Pitman sampling model (EP-SM) is a distribution for random partitions of the set {1,…,n}, with n∈N, which is indexed by real parameters α and θ such that either α∈[0,1) and θ&gt, −α, or α&lt, 0 and θ=−mα for some m∈N. For α=0, the EP-SM is reduced to the Ewens sampling model (E-SM), which admits a well-known compound Poisson perspective in terms of the log-series compound Poisson sampling model (LS-CPSM). In this paper, we consider a generalisation of the LS-CPSM, referred to as the negative Binomial compound Poisson sampling model (NB-CPSM), and we show that it leads to an extension of the compound Poisson perspective of the E-SM to the more general EP-SM for either α∈(0,1), or α&lt, 0. The interplay between the NB-CPSM and the EP-SM is then applied to the study of the large n asymptotic behaviour of the number of blocks in the corresponding random partitions—leading to a new proof of Pitman’s α diversity. We discuss the proposed results and conjecture that analogous compound Poisson representations may hold for the class of α-stable Poisson–Kingman sampling models—of which the EP-SM is a noteworthy special case.

The energy E(G) of a graph G is the sum of the absolute values of its adjacency matrix. In contrast, the trace norm of a digraph Q, which is the sum of the singular values of the corresponding adjacency matrix, is the oriented version of the energy of a graph. It is worth pointing out that one of the main problems in this theory consists of determining appropriated bounds of these types of energies for significant classes of graphs, digraphs and matrices, provided that, in general, finding out their exact values is a problem of great difficulty. In this paper, the trace norm of a {0,1}-Brauer configuration is introduced. It is estimated and computed by associating suitable families of graphs and posets to Brauer configuration algebras.

The order of appearance of n (in the Fibonacci sequence) z(n) is defined as the smallest positive integer k for which n divides the k—the Fibonacci number Fk. Very recently, Trojovský proved that z(n) is an even number for almost all positive integers n (in the natural density sense). Moreover, he conjectured that the same is valid for the set of integers n≥1 for which the integer 4 divides z(n). In this paper, among other things, we prove that for any k≥1, the number z(n) is divisible by 2k for almost all positive integers n (in particular, we confirm Trojovský’s conjecture).

A generalization of the well-known Fibonacci sequence is the k−Fibonacci sequence whose first k terms are 0,…,0,1 and each term afterwards is the sum of the preceding k terms. In this paper, we find all k-Fibonacci numbers that are curious numbers (i.e., numbers whose base ten representation have the form a⋯ab⋯ba⋯a). This work continues and extends the prior result of Trojovský, who found all Fibonacci numbers with a prescribed block of digits, and the result of Alahmadi et al., who searched for k-Fibonacci numbers, which are concatenation of two repdigits.

It is easily observed that the vertices of a simple graph cannot have pairwise distinct degrees. This means that no simple graph of the order of at least two is, in this way, irregular. However, a multigraph can be irregular. Chartrand et al., in 1988, posed the following problem: in a loopless multigraph, how can one determine the fewest parallel edges required to ensure that all vertices have distinct degrees? This problem is known as the graph labeling problem and, for its solution, Chartrand et al. introduced irregular assignments. The irregularity strength of a graph G is known as the maximal edge label used in an irregular assignment, minimized over all irregular assignments. Thus, the irregularity strength of a simple graph G is equal to the smallest maximum multiplicity of an edge of G in order to create an irregular multigraph from G. In the present paper, we show the existence of a required irregular labeling scheme that proves the exact value of the irregularity strength of wheels. Then, we modify this irregular mapping in six cases and obtain labelings that determine the exact value of the modular irregularity strength of wheels as a natural modification of the irregularity strength.

Countably recognizable group classes were introduced by Reinhold Baer and provide a very ingenious way to study large groups through the properties of their countable subgroups. This is the reason we have chosen the countable recognizability to start this series of survey papers on infinite group theory.

In the paper, we present a study devoted to the utilization of multiple-solution tasks (MSTs) in combinatorics as a part of a pre-service teachers course on didactics of mathematics from the view of the mathematics teachers’ specialized knowledge (MTSK) theoretical framework. The study was carried out over the standard course of a summer semester in 2021. The course was attended by 13 pre-service teachers (PSTs). It was carried out online, due to COVID-19 restrictions. Ten combinatorial multiple-solution tasks were assigned to the PSTs. Analyzing pre-service teachers solutions to these tasks, we sought the description and better understanding of the combinatorial knowledge of the topic from the perspective of MSTK. The results revealed some critical aspects of mathematical knowledge in combinatorics that pre-service teachers education should focus on.

The Estrada index of a graph G is defined as EE(G)=∑i=1neλi, where λ1,λ2,…,λn are the eigenvalues of the adjacency matrix of G. A unicyclic graph is a connected graph with a unique cycle. Let U(n,d) be the set of all unicyclic graphs with n vertices and diameter d. In this paper, we give some transformations which can be used to compare the Estrada indices of two graphs. Using these transformations, we determine the graphs with the maximum Estrada indices among U(n,d). We characterize two candidate graphs with the maximum Estrada index if d is odd and three candidate graphs with the maximum Estrada index if d is even.

A new domination parameter in a fuzzy digraph is proposed to espouse a contribution in the domain of domination in a fuzzy graph and a directed graph. Let GD*=V,A be a directed simple graph, where V is a finite nonempty set and A=x,y:x,y∈V,x≠y. A fuzzy digraph GD=σD,μD is a pair of two functions σD:V→0,1 and μD:A→0,1, such that μDx,y≤σDx∧σDy, where x,y∈V. An edge μDx,y of a fuzzy digraph is called an effective edge if μDx,y=σDx∧σDy. Let x,y∈V. The vertex σDx dominates σDy in GD if μDx,y is an effective edge. Let S⊆V, u∈V\S, and v∈S. A subset σDS⊆σD is a dominating set of GD if, for every σDu∈σD\σDS, there exists σDv∈σDS, such that σDv dominates σDu. The minimum dominating set of a fuzzy digraph GD is called the domination number of a fuzzy digraph and is denoted by γGD. In this paper, the concept of domination in a fuzzy digraph is introduced, the domination number of a fuzzy digraph is characterized, and the domination number of a fuzzy dipath and a fuzzy dicycle is modeled.

In this paper, we introduce and investigate an ideal-based dot total graph of commutative ring R with nonzero unity. We show that this graph is connected and has a small diameter of at most two. Furthermore, its vertex set is divided into three disjoint subsets of R. After that, connectivity, clique number, and girth have also been studied. Finally, we determine the cases when it is Eulerian, Hamiltonian, and contains a Eulerian trail.

A Cayley graph Γ=Cay(G,S) is said to be normal if the base group G is normal in AutΓ. The concept of the normality of Cayley graphs was first proposed by M.Y. Xu in 1998 and it plays a vital role in determining the full automorphism groups of Cayley graphs. In this paper, we construct an example of a 2-arc transitive hexavalent nonnormal Cayley graph on the alternating group A119. Furthermore, we determine the full automorphism group of this graph and show that it is isomorphic to A120.

Let (Fn)n be the sequence of Fibonacci numbers. The order of appearance (in the Fibonacci sequence) of a positive integer n is defined as z(n)=min{k≥1:n∣Fk}. Very recently, Trojovská and Venkatachalam proved that, for any k≥1, the number z(n) is divisible by 2k, for almost all integers n≥1 (in the sense of natural density). Moreover, they posed a conjecture that implies that the same is true upon replacing 2k by any integer m≥1. In this paper, in particular, we prove this conjecture.

A graph G is said to be k-γc-critical if the connected domination number γc(G) is equal to k and γc(G+uv)&lt, k for any pair of non-adjacent vertices u and v of G. Let ζ be the number of cut vertices of G and let ζ0 be the maximum number of cut vertices that can be contained in one block. For an integer ℓ≥0, a graph G is ℓ-factor critical if G−S has a perfect matching for any subset S of vertices of size ℓ. It was proved by Ananchuen in 2007 for k=3, Kaemawichanurat and Ananchuen in 2010 for k=4 and by Kaemawichanurat and Ananchuen in 2020 for k≥5 that every k-γc-critical graph has at most k−2 cut vertices and the graphs with maximum number of cut vertices were characterized. In 2020, Kaemawichanurat and Ananchuen proved further that, for k≥4, every k-γc-critical graphs satisfies the inequality ζ0(G)≤mink+23,ζ. In this paper, we characterize all k-γc-critical graphs having k−3 cut vertices. Further, we establish realizability that, for given k≥4, 2≤ζ≤k−2 and 2≤ζ0≤mink+23,ζ, there exists a k-γc-critical graph with ζ cut vertices having a block which contains ζ0 cut vertices. Finally, we proved that every k-γc-critical graph of odd order with minimum degree two is 1-factor critical if and only if 1≤k≤2. Further, we proved that every k-γc-critical K1,3-free graph of even order with minimum degree three is 2-factor critical if and only if 1≤k≤2.

We study the asymptotic behavior in a neighborhood of zero of the sum of a sine series g(b,x)=∑k=1∞bksinkx whose coefficients constitute a convex slowly varying sequence b. The main term of the asymptotics of the sum of such a series was obtained by Aljančić, Bojanić, and Tomić. To estimate the deviation of g(b,x) from the main term of its asymptotics bm(x)/x, m(x)=[π/x], Telyakovskiĭ used the piecewise-continuous function σ(b,x)=x∑k=1m(x)−1k2(bk−bk+1). He showed that the difference g(b,x)−bm(x)/x in some neighborhood of zero admits a two-sided estimate in terms of the function σ(b,x) with absolute constants independent of b. Earlier, the author found the sharp values of these constants. In the present paper, the asymptotics of the function g(b,x) on the class of convex slowly varying sequences in the regular case is obtained.