ROLITO EBALLE

Central Mindanao University, Mathematics Department, Faculty Member

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A tennis enthusiast who is working in some domination-related parameters and centrality measures in graph theory..
Supervisors: DR. SERGIO R. CANOY, JR. and DR. FE ANNABEL N. YEBRON

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Mindanao State University - Iligan Institute of Technology

Hebert Pérez Rosés

Universitat Rovira i Virgili

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Mindanao State University - Iligan Institute of Technology

Michael Henning

Cris Armada

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Papers by ROLITO EBALLE

Independent Semitotal Domination in the Corona of Graphs

Advances and Applications in Discrete Mathematics

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Another Look of Rings Domination in Ladder Graph

Asian research journal of mathematics, Dec 6, 2022

For a nontrivial connected graph G with no isolated vertex, a nonempty subset D ⊆ V (G) is a ring... more For a nontrivial connected graph G with no isolated vertex, a nonempty subset D ⊆ V (G) is a rings dominating set if each vertex v ∈ V − D is adjacent to at least two vertices in V − D. Thus, the dominating set D of V (G) is a rings dominating set if for all v ∈ V − D, |N (v) ∩ (V − D)| ≥ 2. The cardinality of minimum rings dominating set of G is the rings domination number of G, denoted by γri whereas the cardinality of maximum rings dominating set is the upper rings domination number and is denoted by γ ri. Here, we determine how the rings dominating set is constructed in the ladder graph with the inclusion of generated conditions for this type of domination and give new approach for its parameter.

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Domination Defect in the Edge Corona of Graphs

Asian research journal of mathematics, Dec 19, 2022

Given a graph G = (V (G), E(G)), a nonempty set S ⊆ V (G) of fixed cardinality γ(G) − k is called... more Given a graph G = (V (G), E(G)), a nonempty set S ⊆ V (G) of fixed cardinality γ(G) − k is called a ζ k − set of G, where 1 ≤ k ≤ γ(G) − 1, if S gives the minimum cardinality |V (G) \ NG[S]| for all the possible subsets of V (G), each of which has γ(G) − k elements. This is the number of vertices in G which are left undominated by S. In this paper, the k-domination defects of graphs resulting from the binary operation edge corona G H are characterized and as a direct consequence, the corresponding k-domination defect ζ k (G H) is then determined.

Restrained Weakly Connected 2-Domination in the Join of Graphs

Communications in Mathematics and Applications

Let G = (V (G), E(G)) be a connected graph. A restrained weakly connected 2-dominating (RWC2D) se... more Let G = (V (G), E(G)) be a connected graph. A restrained weakly connected 2-dominating (RWC2D) set in G is a subset D ⊆ V (G) such that every vertex in V (G)\D is dominated by at least two vertices in D and is adjacent to a vertex in V (G)\D, and that the subgraph 〈D〉 w weakly induced by D is connected. The restrained weakly connected 2-domination number of G, denoted by γ r2w (G), is the smallest cardinality of a restrained weakly connected 2-dominating set in G. In this paper, we characterize the RWC2D sets in the join of two graphs G and H, each of which is of order at least 3 and has no isolated vertex, and in the join K 1 ∨ F , where K 1 is the trivial graph and that at least one component of F is of order at least 3. In particular, it is shown that 2 ≤ γ r2w (G ∨ H) ≤ 4 and γ r2w (K 1 ∨ F) = min{1 + γ r (F), γ 2 (F)}, where γ r and γ 2 are the restrained domination and 2-domination parameters, respectively.

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Clique Centrality and Global Clique Centrality of Graphs

Asian Research Journal of Mathematics

We formally introduce in this paper two parameters in graph theory, namely, clique centrality and... more We formally introduce in this paper two parameters in graph theory, namely, clique centrality and global clique centrality. Let G be a finite, simple and undirected graph of order n. A clique in G is a nonempty subset W \(\subseteq\) V (G) such that the subgraph \(\langle\)W\(\rangle\)G induced by W is complete. The maximum size of any clique containing vertex u \(\in\) V (G) is called the clique centrality of u in G. Normalizing the sum of the clique centralities of all the vertices of G will lead us to the global clique centrality of G, whose value ranges from \(\frac{1}{m}\) to 1. In this paper, we study some general properties of the global clique centrality and then evaluate it for some parameterized families of graphs.

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Introducing the Clique-Safe Domination in Graphs

Asian Research Journal of Mathematics

Let G = (V (G), E(G)) be any finite, undirected, simple graph. A set D \(\subseteq\) V (G) is int... more Let G = (V (G), E(G)) be any finite, undirected, simple graph. A set D \(\subseteq\) V (G) is introduced in this paper as a clique-safe dominating set of G if D is a dominating set of G and for every clique D\(\prime\)m of size m in the subgraph induced by V (G) \D, there exists a clique Dn of size n in the subgraph induced by D such that n \(\ge\) m. The clique-safe domination number of G, denoted by \(\gamma\)cs (G) is the smallest cardinality of a clique-safe dominating set of G. This study aims to generate a few elementary properties of the parameter and to characterize the minimum clique-safe dominating sets of paths and cycles. As a consequence, the clique-safe domination numbers of the aforesaid graphs are obtained.

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Independent Semitotal Domination in the Join of Graphs

Asian Research Journal of Mathematics

A subset W \(\subseteq\) V (G) of a graph G is an independent semitotal dominating set of G, abbr... more A subset W \(\subseteq\) V (G) of a graph G is an independent semitotal dominating set of G, abbreviated ISTd-set of G, if W is an independent dominating set of G and every element of W is exactly of distance 2 from at least one other element of W. The independent semitotal domination number of G, denoted by \(\gamma\)it2(G), is the minimum cardinality of an ISTd-set of G. In this paper, we study the concept of independent semitotal domination in graphs and investigate the conditions for graphs on which the ISTd-sets exist. Further, the ISTd-sets of the join of any two graphs are examined. Consequently, the corresponding independent semitotal domination number of these graphs are obtained.

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Global Clustering Coeffcient of the Join and Corona of Graphs

Asian Research Journal of Mathematics

The global clustering coecient is one of the most useful indices in complex network analysis. It ... more The global clustering coecient is one of the most useful indices in complex network analysis. It is another metric that somehow measures how close a graph from being a complete graph. In this paper we present some expressions for the global clustering coeffcient of the join G v H and corona G o H of arbitrary simple and undirected graphs G and H. As corollaries to these results, we will show that for the path Pm, cycle Cm, fan Fm, and wheel Wm, both Cc(Pm v Pm) and Cc(Cm v Cm) approach to 0 as m increases without bound, while Cc(Fm), Cc(Wm), Cc(Pm o Pm), and Cc(Cm o Cm) all approach to 2/3.

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Weakly connected 2-domination in the lexicographic product of graphs

International Journal of Mathematical Analysis, 2022

Harmonic Centralization of Some Graph Families

Advances and applications in discrete mathematics, May 2, 2022

Centrality describes the importance of nodes in a graph and is modeled by various measures. Its g... more Centrality describes the importance of nodes in a graph and is modeled by various measures. Its global analogue, called centralization, is a general formula for calculating a graph-level centrality score based on the node-level centrality measure. The latter enables us to compare graphs based on the extent to which the connections of a given network are concentrated on a single vertex or

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Global Clustering Coefficient of the Products of Complete Graphs

Asian research journal of mathematics, May 23, 2022

Clustering Coefficient of the Tensor Product of Graphs

Asian Research Journal of Mathematics

Clustering coefficient is one of the most useful indices in complex networks. However, graph theo... more Clustering coefficient is one of the most useful indices in complex networks. However, graph theoretic properties of this metric have not been discussed much in the literature, especially in graphs resulting from some binary operations. In this paper we present some expressions for the clustering coefficient of the tensor product of arbitrary graphs, regular graphs, and strongly regular graphs. A Vizing-type upperbound and a sharp lower bound for the clustering coefficient of the tensor product of graphs are also given.

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Harmonic Centrality and Centralization of Some Graph Products

Asian Research Journal of Mathematics

Harmonic centrality calculates the importance of a node in a network by adding the inverse of the... more Harmonic centrality calculates the importance of a node in a network by adding the inverse of the geodesic distances of this node to all the other nodes. Harmonic centralization, on the other hand, is the graph-level centrality score based on the node-level harmonic centrality. In this paper, we present some results on both the harmonic centrality and harmonic centralization of graphs resulting from some graph products such as Cartesian and direct products of the path P2 with any of the path Pm, cycle Cm, and fan Fm graphs.

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Harmonic Centralization of Some Graph Families

ArXiv, 2022

Centrality describes the importance of nodes in a graph and is modeled by various measures. Freem... more Centrality describes the importance of nodes in a graph and is modeled by various measures. Freeman’s centralization, on the other hand, is a general method for calculating a graph-level centrality score based on a node-level centrality measure. The latter enables us to compare graphs based on the extent to which the connections of a given network are concentrated on a single vertex or group of vertices. One of the measures of centrality in social network analysis is harmonic centrality. It sums the inverse of the geodesic distances of each node to other nodes where it is 0 if there is no path from one node to another, with the sum normalized by dividing it by m − 1, where m is the number of nodes of the graph. In this paper, we present some results regarding the harmonic centralization of some important families of graphs with the hope that formulas generated herein will be of use when one determines the harmonic centralization of more complex graphs.

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Harmonic Centrality of Some Graph Families

ArXiv, 2021

One of the more recent measures of centrality in social network analysis is the normalized harmon... more One of the more recent measures of centrality in social network analysis is the normalized harmonic centrality. A variant of the closeness centrality, harmonic centrality sums the inverse of the geodesic distances of each node to other nodes where it is 0 if there is no path from one node to another. It is then normalized by dividing it by m − 1, where m is the number of nodes of the graph. In this paper, we present notions regarding the harmonic centrality of some important classes of graphs.

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Exploring the vertex and edge corona of graphs for their weakly connected 2-domination

by Mae Militante and ROLITO EBALLE

International Journal of Contemporary Mathematical Sciences, 2021

This article is distributed under the Creative Commons by-nc-nd Attribution License.

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Weakly connected 2-domination in some special graphs

Applied Mathematical Sciences

This article is distributed under the Creative Commons by-nc-nd Attribution License.

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Domination defect for the join and corona of graphs

The domination number of a graph G denoted by γ(G) is the minimum number of vertices required to ... more The domination number of a graph G denoted by γ(G) is the minimum number of vertices required to dominate all the vertices of G. The minimality of γ(G) implies that if W ⊆ V (G) such that |W | < γ(G), then there is at least one vertex of G that is not dominated byW . Given a positive integer k < γ(G), where γ(G) ≥ 2, the k-domination defect of G denoted by ζk(G) is the minimum number of vertices of G left undominated by any subset of vertices of G with cardinality γ(G)− k. In this paper, we study further the concept of k-domination defect of a graph G and investigate it for graphs resulting from some binary operations. In particular, the k-domination defect sets of the join and corona of two nontrivial graphs are examined. Consequently, the corresponding k-domination defect of these graphs are obtained. Mathematics Subject Classification: 05C69, 05C76

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m-Convex hulls in graphs resulting from some binary operations

Let G be a connected graph u and v be two vertices in V (G). The set JG[u, v] denotes the closed ... more Let G be a connected graph u and v be two vertices in V (G). The set JG[u, v] denotes the closed interval consisting of u, v and all vertices lying on some u-v m-path in G. A subset C of V (G) is m-convex if JG[u, v] ⊆ C for every pair of vertices u, v ∈ C. The m-convex hull of a subset S of V (G), denoted by m[S]G, is the smallest m-convex set in G containing S. This study determines the m-convex hulls of subsets in the join, composition and cartesian product of two graphs. Mathematics Subject Classification: 05C12

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Global Defensive Alliances in the Join, Corona and Composition of Graphs

ABSTRACT By a defensive alliance in a graph G we mean any set S of vertices in G such that each v... more ABSTRACT By a defensive alliance in a graph G we mean any set S of vertices in G such that each vertex in S is adjacent to at least as many vertices inside S, including the vertex itself, as outside S. If, in addition, we require that every vertex outside a defensive alliance S is adjacent to at least one vertex in S, then S becomes a global defensive alliance. The minimum cardinality of a global defensive alliance is the global defensive alliance number of G. In this paper, we determine bounds for the global defensive alliance numbers in the join, corona and composition of graphs.

Independent Semitotal Domination in the Corona of Graphs

Advances and Applications in Discrete Mathematics

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Another Look of Rings Domination in Ladder Graph

Asian research journal of mathematics, Dec 6, 2022

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Domination Defect in the Edge Corona of Graphs

Asian research journal of mathematics, Dec 19, 2022