ON HOP DOMINATION NUMBER OF SOME GENERALIZED GRAPH STRUCTURES

A subset \( H \subseteq V (G) \) of a graph \(G\) is a hop dominating set (HDS) if for every \({v\in (V\setminus H)}\) there is at least one vertex \(u\in H\) such that \(d(u,v)=2\). The minimum cardinality of a hop dominating set of \(G\) is called the hop domination number of \(G\) and is denoted by \(\gamma_{h}(G)\). In this paper, we compute the hop domination number for triangular and quadrilateral snakes. Also, we analyse the hop domination number of graph families such as generalized thorn path, generalized ciliates graphs, glued path graphs and generalized theta graphs.

UNIT AND UNITARY CAYLEY GRAPHS FOR THE RING OF EISENSTEIN INTEGERS MODULO \(n\)

Let \({E}_{n}\) be the ring of Eisenstein integers modulo \(n\). We denote by \(G({E}_{n})\) and \(G_{{E}_{n}}\), the unit graph and the unitary Cayley graph of \({E}_{n}\), respectively. In this paper, we obtain the value of the diameter, the girth, the clique number and the chromatic number of these graphs. We also prove that for each \(n>1\), the graphs \(G(E_{n})\) and \(G_{E_{n}}\) are Hamiltonian.

A ROBUST ITERATIVE APPROACH FOR SOLVING NONLINEAR VOLTERRA DELAY INTEGRO–DIFFERENTIAL EQUATIONS

This paper presents a new iterative algorithm for approximating the fixed points of multivalued generalized \(\alpha\)–nonexpansive mappings. We study the stability result of our new iterative algorithm for a larger concept of stability known as weak \(w^2\)–stability. Weak and strong convergence results of the proposed iterative algorithm are also established. Furthermore, we show numerically that our new iterative algorithm outperforms several known iterative algorithms for multivalued generalized \(\alpha\)–nonexpansive mappings. Again, as an application, we use our proposed iterative algorithm to find the solution of nonlinear Volterra delay integro-differential equations. Finally, we provide an illustrative example to validate the mild conditions used in the result of the application part of this study. Our results improve, generalize and unify several results in the existing literature.

DIFFERENTIAL GAME WITH A LIFELINE FOR THE INERTIAL MOVEMENTS OF PLAYERS

In this paper, we study the well-known problem of Isaacs called the "Life line" game when movements of players occur by acceleration vectors, that is, by inertia in Euclidean space. To solve this problem, we investigate the dynamics of the attainability domain of an evader through finding solvability conditions of the pursuit-evasion problems in favor of a pursuer or an evader. Here a pursuit problem is solved by a parallel pursuit strategy. To solve an evasion problem, we propose a strategy for the evader and show that the evasion is possible from given initial positions of players. Note that this work develops and continues studies of Isaacs, Petrosjan, Pshenichnii, Azamov, and others performed for the case of players' movements without inertia.

PRODUCTS OF ULTRAFILTERS AND MAXIMAL LINKED SYSTEMS ON WIDELY UNDERSTOOD MEASURABLE SPACES

Constructions related to products of maximal linked systems (MLSs) and MLSs on the product of widely understood measurable spaces are considered (these measurable spaces are defined as sets equipped with \(\pi\)-systems of their subsets; a \(\pi\)-system is a family closed with respect to finite intersections). We compare families of MLSs on initial spaces and MLSs on the product. Separately, we consider the case of ultrafilters. Equipping set-products with topologies, we use the box-topology and the Tychonoff product of Stone-type topologies. The properties of compaction and homeomorphism hold, respectively.

SHILLA GRAPHS WITH \(b=5\) AND \(b=6\)

A \(Q\)-polynomial Shilla graph with \(b = 5\) has intersection arrays \(\{105t,4(21t+1),16(t+1); 1,4 (t+1),84t\}\), \(t\in\{3,4,19\}\). The paper proves that distance-regular graphs with these intersection arrays do not exist. Moreover, feasible intersection arrays of \(Q\)-polynomial Shilla graphs with \(b = 6\) are found.

LINEARIZATION OF POISSON–LIE STRUCTURES ON THE 2D EUCLIDEAN AND (1 + 1) POINCARÉ GROUPS

The paper deals with linearization problem of Poisson-Lie structures on the \((1+1)\) Poincaré and \(2D\) Euclidean groups. We construct the explicit form of linearizing coordinates of all these Poisson-Lie structures. For this, we calculate all Poisson-Lie structures on these two groups mentioned above, through the correspondence with Lie Bialgebra structures on their Lie algebras which we first determine.

ON A CLASS OF EDGE-TRANSITIVE DISTANCE-REGULAR ANTIPODAL COVERS OF COMPLETE GRAPHS

The paper is devoted to the problem of classification of edge-transitive distance-regular antipodal covers of complete graphs. This extends the classification of those covers that are arc-transitive, which has been settled except for some tricky cases that remain to be considered, including the case of covers satisfying condition \(c_2=1\) (which means that every two vertices at distance 2 have exactly one common neighbour).Here it is shown that an edge-transitive distance-regular antipodal cover of a complete graph with \(c_2=1\) is either the second neighbourhood of a vertex in a Moore graph of valency 3 or 7, or a Mathon graph, or a half-transitive graph whose automorphism group induces an affine \(2\)-homogeneous group on the set of its fibres. Moreover, distance-regular antipodal covers of complete graphs with \(c_2=1\) that admit an automorphism group acting \(2\)-homogeneously on the set of fibres (which turns out to be an approximation of the property of edge-transitivity of such cover), are described. A well-known correspondence between distance-regular antipodal covers of complete graphs with \(c_2=1\) and geodetic graphs of diameter two that can be viewed as underlying graphs of certain Moore geometries, allows us to effectively restrict admissible automorphism groups of covers under consideration by combining Kantor's classification of involutory automorphisms of these geometries together with the classification of finite 2-homogeneous permutation groups.

CARLEMAN'S FORMULA OF A SOLUTIONS OF THE POISSON EQUATION IN BOUNDED DOMAIN

We suggest an explicit continuation formula for a solution to the Cauchy problem for the Poisson equation in a domain from its values and values of its normal derivative on a part of the boundary. We construct the continuation formula of this problem based on the Carleman--Yarmuhamedov function method.

NOTE ON SUPER \((a,1)\)–\(P_3\)–ANTIMAGIC TOTAL LABELING OF STAR \(S_n\)

Let \(G=(V, E)\) be a simple graph and \(H\) be a subgraph of \(G\). Then \(G\) admits an \(H\)-covering, if every edge in \(E(G)\) belongs to at least one subgraph of \(G\) that is isomorphic to \(H\). An \((a,d)-H\)-antimagic total labeling of \(G\) is bijection \(f:V(G)\cup E(G)\rightarrow \{1, 2, 3,\dots, |V(G)| + |E(G)|\}\) such that for all subgraphs \(H'\) of \(G\) isomorphic to \(H\), the \(H'\) weights \(w(H') =\sum_{v\in V(H')} f (v) + \sum_{e\in E(H')} f (e)\) constitute an arithmetic progression \(\{a, a + d, a + 2d, \dots , a + (n- 1)d\}\), where \(a\) and \(d\) are positive integers and \(n\) is the number of subgraphs of \(G\) isomorphic to \(H\). The labeling \(f\) is called a super \((a, d)-H\)-antimagic total labeling if \(f(V(G))=\{1, 2, 3,\dots, |V(G)|\}.\) In [5], David Laurence and Kathiresan posed a problem that characterizes the super \( (a, 1)-P_{3}\)-antimagic total labeling of Star \(S_{n},\) where \(n=6,7,8,9.\) In this paper, we completely solved this problem.