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  • Three new classes of binomial Fibonacci sums
    Robert Frontczak *
    In this paper, we introduce three new classes of binomial sums involving Fibonacci (Lucas) numbers and weighted binomial coefficients. One particular result is linked to a problem proposal recently published in the journal The Fibonacci Quarterly.
    Keywords: Binomial Coefficient, Fibonacci Number, Lucas Number
  • Mahdieh Azari *
    The transmission of a vertex ${\varsigma}$ in a connected graph $\mathcal{J}$ is the sum of distances between ${\varsigma}$ and all other vertices of $\mathcal{J}$. A graph $\mathcal{J}$ is called transmission regular if all vertices have the same transmission. In this paper, we propose a new graph invariant for measuring the transmission irregularity extent of transmission irregular graphs. This invariant which we call the total transmission irregularity number (TTI number for short) is defined as the sum of the absolute values of the difference of the vertex transmissions over all unordered vertex pairs of a graph. We investigate some lower and upper bounds on the TTI number which reveal its connection to a number of already established indices. In addition, we compute the TTI number for various families of composite graphs and for some chemical graphs and nanostructures derived from them.
    Keywords: Transmission Of A Vertex, Transmission Irregular Graph, Graph Invariant, Bound, Composite Graph
  • Javad Tayyebi *, Hassan Hassanpour, Hamid Bigdeli, Elham Hosseinzadeh
    This paper addresses a bimatrix game in which the satisfactory degrees of the players are vague. Type-2 fuzzy goal programming technique is used to describe the game. Then, the notion of equilibrium points is introduced and an optimization problem is given to calculate them. Moreover, the special case that the type-2 fuzzy goals are triangular is investigated. Finally, an applicable example is presented to illustrate the results.
    Keywords: Bimatrix Games, Type-2 Fuzzy Goals, Equilibrium Points, Triangular Type-2 Fuzzy Numbers
  • Rikio Ichishima *, Francisco Antonio Muntaner-Batle, Yukio Takahashi
    A numbering $f$ of a graph $G$ of order $n$ is a labeling that assigns distinct elements of the set $\left\{1,2, \ldots, n \right\}$ to the vertices of $G$. The strength $\mathrm{str}\left(G\right) $ of $G$ is defined by $\mathrm{str}\left( G\right) =\min \left\{ \mathrm{str}_{f}\left( G\right)\left\vert f\text{ is a numbering of }G\right. \right\}$, where $\mathrm{str}_{f}\left( G\right) =\max \left\{ f\left( u\right)+f\left( v\right) \left\vert uv\in E\left( G\right) \right. \right\} $.Using the concept of independence number of a graph, we determine formulas for the strength of powers of paths and cycles. To achieve the latter result, we establish a sharp upper bound for the strength of a graph in terms of its order and independence number and a formula for the independence number of powers of cycles.
    Keywords: Strength, Independence Number, $K$Th Power Of A Graph, Graph Labeling, Combinatorial Optimization
  • Yeni Susanti *, Muhammad Huda, Ramadhani Firmansyah
    In the context of a finite undirected graph $\zeta$, an edge irregular labelling is defined as a labelling of its vertices with some labels in such a way that each edge has a unique weight, which is determined by the sum of the labels of its endpoints. The main objective of this study is to determine the smallest positive integer $n$ for which it is possible to assign a total edge irregular labelling to $\zeta$ with $n$ as the biggest label. This investigation focuses particularly on cases where $\zeta$ represents the generalized arithmetic and generalized geometric staircase graphs. Within this paper, the definition of generalized geometric staircase graph is proposed. Moreover, we not only establish the edge irregularity strength of these two kind of graphs but also present a method for creating the corresponding edge irregular labelling.
    Keywords: Irregular Labeling, Staircase Graphs, Total Edge Irregularity Strength
  • Anand Tiwari *, Yogendra Singh, Amit Tripathi
    The $k$-semi equivelar maps, for $k \geq 2$, are generalizations of maps on the surfaces of Johnson solids to closed surfaces other than the 2-sphere. In the present study, we determine 2-semi equivelar maps of curvature 0 exhaustively on the torus and the Klein bottle. Furthermore, we classify (up to isomorphism) all these 2-semi equivelar maps on the surfaces with up to 12 vertices.
    Keywords: 2-Semi Equivelar Maps, Face-Sequence, Torus, Klein Bottle
  • Michal Staš *, Maria Timková
    The crossing number $\mathrm{cr}(G)$ of a graph $G$ is the minimum number of edge crossings over all drawings of $G$ in the plane. In the paper, we extend known results concerning crossing numbers of join products of two small graphs with cycles. The crossing number of the join product $G^\ast + C_n$ for the disconnected graph $G^\ast$ consisting of the complete graph $K_{4}$ and one isolated vertex is given, where $C_n$ is the cycle on $n$ vertices. The proof of the main result is done with the help of lemma whose proof is based on a special redrawing technique. Up to now, the crossing numbers of $G + C_n$ were done only for a few disconnected graphs $G$. Finally, by adding new edge to the graph $G^\ast$, we are able to obtain the crossing number of $G_1+C_n$ for one other graph $G_1$ of order five.
    Keywords: Graph, Crossing Number, Join Product, Separating Cycle, Cycle
  • On twin EP numbers
    Ömer Eğecioğlu *, Bünyamin Şahin

    EP numbers were introduced by Estrada and Pogliani in 2008. These are positive integers $E(n)$ defined as the product of $n$ and the sum of the digits of $n$. Estrada and Pogliani suspected that there may be infinitely many twin EP numbers; i.e. those pairs in this sequence that differ by one. It is relatively easy to show that three consecutive EP numbers do not exist, and that no pair $E(n)$ and $E(m)$ can be twins for infinitely many bases $b$. The main contribution of our work is the result that indeed there are infinitely many twin EP numbers over any base. The proof is constructive and makes use of elementary properties of natural numbers. The forms of the twin EP numbers presented are derived from continued fractions. The behavior of the series of the reciprocals of twin EP numbers is also considered.

    Keywords: EP Number, Twin EP Number, Continued Fractions, Reciprocal Series
  • $G$-designs for the connected triangular bicyclic graphs with nine edges
    Bryan Freyberg *, Dalibor Froncek, Joel Jeffries, Gretta Jensen, Andrew Sailstad
    A $G$-design of order $n$ is a decomposition of the complete graph $K_n$ into isomorphic copies of $G$. We show that if $G$ is a connected bicyclic graph with nine edges containing two triangles, a $G$-design of order $n$ exists whenever $n \equiv 0,1 \pmod{18}$.
    Keywords: Graph Decomposition, Bicyclic Graphs, Rho-Labelings
  • Jafar Pourmahmoud *, Davoud Norouzi Bene
    Data Envelopment Analysis measures relative efficiency, in which the performances of the DMUs in a group are compared. In this approach, an efficient unit in one group may be considered inefficient compared to the units of other groups and vice versa. To solve this weakness, two known productivity indexes, the Malmquist and Luenberger, have been introduced to evaluate units (or systems) from one period to another. The existence of special types of data such as undesirable and non-discretionary in some multi-stage series systems is unavoidable. The evaluation of such systems in the simultaneous presence of the mentioned data and different periods has not been done so far. Therefore, in this study, we have presented a model with a new approach to evaluate them. At the end of the study, we checked the proposed model’s ability by providing comparative and structural examples. We have shown that without undesirable and non-discretionary data, the proposed is better than other models. Also, this model has been used for the first time and obtained acceptable results in the presence of these data.
    Keywords: Network Data Envelopment Analysis, Malmquist Productivity Index, Evaluation, Non-Discretionary Data, Undesirable Data
  • Star-critical connected Ramsey numbers for 2-colorings of complete graphs
    Monu Moun, Jagjeet Jakhar, Mark Budden *
    This paper builds upon Sumner's work by further investigating the concept of connected Ramsey numbers, specifically focusing on star-critical connected Ramsey numbers. We obtain star-critical connected Ramsey numbers for several cases of trees versus complete graphs, stars versus stars, and paths versus paths. The connected Ramsey number for a star versus $K_3$ is also evaluated. Exact values are also obtained for the connected Ramsey numbers of $K_{1,n}$ versus $K_3$. This research explores the interplay between connectivity and graph coloring within the context of Ramsey theory.
    Keywords: Ramsey Number, Connected Ramsey Number, Star-Critical Ramsey Number
  • Jesmina Pervin, Lavanya Selvaganesh *
    A graph $G$ is said to be $H$-free if $G$ does not contain $H$ as an induced subgraph. Let $\mathcal{S}_{n}^2(m)$ be a \textit{variation of double star $\mathcal{S}_{n}^2$} obtained by adding m (<=n) disjoint edges between the pendant vertices which are at distance 3 in $\mathcal{S}_{n}^2$. A graph having integer eigenvalues for its signless Laplacian matrix is known as a Q-integral graph. The Q-spectral radius of a graph is the largest eigenvalue of its signless Laplacian. Any connected Q-integral graph G with Q-spectral radius 7 and maximum edge-degree 8 is either $K_{1,4}\square K_2$ or contains $\mathcal{S}_{4}^2(0)$ as an induced subgraph or is a bipartite graph having at least one of the induced subgraphs $\mathcal{S}_{4}^2(m)$, (m=1, 2, 3). In this article, we improve this result by showing that every connected Q-integral graph G having Q-spectral radius 7, maximum edge-degree 8 is always bipartite and $\mathcal{S}_{4}^2(3)$-free.
    Keywords: Edge-Degree, H-Free Graph, Signless Laplacian Matrix, Q-Integral Graph
  • Sasmita Rout, Pawan Mishra, Gautam Das *
    Let $G=(V,E)$ be a simple, undirected and connected graph. A Roman dominating function (RDF) on the graph $G$ is a function $f:V\rightarrow\{0,1,2\}$ such that each vertex $v\in V$ with $f(v)=0$ is adjacent to at least one vertex $u\in V$ with $f(u)=2$. A total Roman dominating function (TRDF) of $G$ is a function $f:V\rightarrow\{0,1,2\}$ such that $(i)$ it is a Roman dominating function, and  $(ii)$ the vertices with non-zero weights induce a subgraph with no isolated vertex. The total Roman dominating set (TRDS) problem is to minimize the associated weight, $f(V)=\sum_{u\in V} f(u)$, called the total Roman domination number ($\gamma_{tR}(G)$). Similarly, a subset $S\subseteq V$ is said to be a total dominating set (TDS) on the graph $G$ if $(i)$ $S$ is a dominating set of $G$, and $(ii)$  the induced subgraph $G[S]$ does not have any isolated vertex. The objective of the TDS problem is to minimize the cardinality of the TDS of a given graph. The TDS problem is NP-complete for general graphs.  In this paper, we propose a simple $10.5\operatorname{-}$factor approximation algorithm for TRDS problem in UDGs. The running time of the proposed algorithm is $O(|V|\log k)$, where $k$ is the number of vertices with weights $2$. It is an improvement over the best-known $12$-factor approximation algorithm with running time $O(|V|\log k)$ available in the literature. Next, we propose another algorithm for the TDS problem in UDGs, which improves the previously best-known approximation factor from $8$ to $7.79$. The running time of the proposed algorithm is $O(|V|+|E|)$.
    Keywords: Total Domination, Total Roman Domination, Approximation Algorithms, Unit Disk Graphs
  • The orders of subgroup products and coset products
    Shigeru Takamura *
    A sect is a subset of a group given by the product of a finite number of subgroups. It is generally not a direct product nor even a subgroup of the group. For finite groups, the orders of sects are their basic invariants. In this paper we describe properties of the orders of sects, such as divisibility and inequalities, which give constraints on the possible values of the orders of sects. We further consider clans, which are subsets of groups given by products of finite numbers of cosets. We also describe properties of their orders.
    Keywords: Subgroup Product, Coset Product, Group Factorization, Order Inequality, Order Divisibility
  • Faiz Zakaria *, Hicham Benaissa, Othmane Baiz
    The objective of this paper is to examine a model of a thermo-electro-elastic body situated on a semi-insulator foundation. Friction is characterized by Tresca's friction law, and the contact is bilateral. The primary contribution is to derive the weak variational formulation of the model, constituting a system that couples three inclusions where the unknowns are the strain field, the electric field, and the temperature field. Subsequently, we demonstrate the unique solvability of the system, along with the continuous dependence of its solution under consideration. The secondary contribution involves the investigation of an associated optimal control problem, for which we establish the existence and convergence results. The proofs rely on arguments related to monotonicity, compactness, convex analysis, and lower semicontinuity.
    Keywords: Thermo-Electro-Elastic Materials, Variational Inequalities, Stationary Inclusion, Continuous Dependence, Optimal Control
  • A. Saibulla *, P. Roushini Pushpam
    A $(p, q)$-graph $G$ is $(a, d)$-edge antimagic total if there exists a bijection $f$ from $V(G) \cup E(G)$ to $\{1, 2, \dots, p+q\}$ such that for each edge $uv \in E(G)$, the edge weight $\Lambda(uv) = f(u) + f(uv) + f(v)$ forms an arithmetic progression with first term $a > 0$ and common difference $d \geq 0$. An $(a, d)$-edge antimagic total labeling in which the vertex labels are $1, 2, \dots, p$ and edge labels are $p+1, p+2, \dots, p+q$ is called a {\it super} $(a, d)$-{\it edge antimagic total labeling}. Another variant of $(a, d)$-edge antimagic total labeling called as e-super $(a, d)$-edge antimagic total labeling in which the edge labels are $1, 2, \dots, q$ and vertex labels are $q+1, q+2, \dots, q+p$. In this paper, we investigate the  existence of e-super $(a, d)$-edge antimagic total labeling for total graphs of paths, copies of cycles and disjoint union of cycles.
    Keywords: Graph Labeling, Magic Labeling, Antimagic Labeling
  • Tsz Lung Chan, Gee-Choon Lau *, Wai Chee Shiu
    An edge labeling of a graph $G = (V, E)$ is said to be local antimagic if it is a bijection $f:E \to\{1,\ldots ,|E|\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x)\not= f^+(y)$, where the induced vertex label $f^+(x)= \sum f(e)$, with $e$ ranging over all the edges incident to $x$. The local antimagic chromatic number of $G$, denoted by $\chi_{la}(G)$, is the minimum number of distinct induced vertex labels over all local antimagic labelings of $G$. In this paper, we study local antimagic labeling of disjoint unions of stars, paths and cycles whose components need not be identical. Consequently, we completely determined the local antimagic chromatic numbers of disjoint union of 2 stars, paths, and 2-regular graphs with at most one odd order component respectively.
    Keywords: Local Antimagic Labeling, Local Antimagic Chromatic Number, Disconnected Graphs
  • Sumanta Das *, Mridul Kanti Sen, Sunil Kumar Maity
    In this paper, we generalize the concept of Cayley graphs associated to finite groups. The aim of this paper is the characterization of graph theoretic properties of new type of directed graph $\Gamma_P(G;S)$ and algebraic properties of Leavitt path algebra of order prime Cayley graph $O\Gamma(G;S)$, where $G$ is a finite group with a generating set $S$. We show that the Leavitt path algebra of order prime Cayley graph $L_K(O\Gamma(G;S))$ of a non trivial finite group $G$ with any generating set $S$ over a field $K$ is a purely infinite simple ring. Finally, we prove that the Grothendieck group of the Leavitt path algebra $L_K(\Gamma_P(D_n;S))$ is isomorphic to $\mathbb{Z}_{2n-1}$, where $D_n$ is the dihedral group of degree $n$ and $S=\left\{a, b\right\}$ is the generating set of $D_n$.
    Keywords: Group, Directed Cayley Graph, Order Prime Cayley Graph, Grothendieck Group
  • Mark Anthony Tolentino *, Janree Ruark Gatpatan, Timothy Robin Teng
    Neighbor-distinguishing colorings, which are colorings that induce a proper vertex coloring of a graph, have been the focus of different studies in graph theory. One such coloring is the set coloring. For a nontrivial graph $G$, let $c:V(G)\to \mathbb{N}$ and define the neighborhood color set $NC(v)$ of each vertex $v$ as the set containing the colors of all neighbors of $v$. The coloring $c$ is called a set coloring if $NC(u)\neq NC(v)$ for every pair of adjacent vertices $u$ and $v$ of $G$. The minimum number of colors required in a set coloring is called the set chromatic number of $G$ and is denoted by $\chi_s (G)$. In recent years, set colorings have been studied with respect to different graph operations such as join, comb product, middle graph, and total graph. Continuing the theme of these previous works, we aim to investigate set colorings of the Cartesian product of graphs. In this work, we investigate the gap given by $\max\{ \chi_s(G), \chi_s(H) \} - \chi_s(G\ \square\ H)$ for graphs $G$ and $H$. In relation to this objective, we determine the set chromatic numbers of the Cartesian product of some graph families.
    Keywords: Set Coloring, Cartesian Product, Neighbor-Distinguishing Coloring
  • Carlos Espinal, Ivan Gutman, Juan Rada *
    Let $G$ be a simple graph. The elliptic Sombor index of $G$ is defined as $$    ESO(G) = \sum_{uv} \left(d_{u}+ d_{v} \right)\sqrt{d^{2}_{u}+d^{2}_{v}},$$  where $d_{u}$ denotes the degree of the vertex $u$, and the sum runs over the set of edges of $G$. In this paper we solve the extremal value problem of $ESO$ over the set of (connected) chemical graphs and over the set of chemical trees, with equal number of vertices.
    Keywords: Elliptic Sombor Index, Chemical Graph, Vertex-Degree-Based Topological Index
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