• Bulletin of the Korean Mathematical Society
• /
• v.44 no.3
• /
• pp.385-394
• /
• 2007

An upper bound of the basis number of the semi-strong product of cycles with bipartite graphs is given. Also, an example is presented where the bound is achieved.

• Communications of the Korean Mathematical Society
• /
• v.35 no.2
• /
• pp.359-370
• /
• 2020

A graph G is called a well covered graph if every maximal independent set in G is maximum, and co-well covered graph if its complement is a well covered graph. We study some properties of a co-well covered graph and we characterize when the join, the corona product, and cartesian product are co-well covered graphs. Also we characterize when powers of trees and cycles are co-well covered graphs. The line graph of a graph which is co-well covered is also studied.

• Journal of applied mathematics & informatics
• /
• v.34 no.3_4
• /
• pp.179-192
• /
• 2016

In this paper, we introduce product interval-valued fuzzy graphs and prove several results which are analogous to interval-valued fuzzy graphs. We conclude by giving properties for a product interval-valued fuzzy graph.

• Kyungpook Mathematical Journal
• /
• v.58 no.4
• /
• pp.747-759
• /
• 2018

A graph G = (V (G), E(G) of order n = |V (G)| and size m = |E(G)| is said to be odd harmonious if there exists an injection $f:V(G){\rightarrow}\{0,\;1,\;2,\;{\ldots},\;2m-1\}$ such that the induced function $f^*:E(G){\rightarrow}\{1,\;3,\;5,\;{\ldots},\;2m-1\}$ defined by $f^*(uv)=f(u)+f(v)$ is bijection. While a bipartite graph G with partite sets A and B is said to be bigraceful if there exist a pair of injective functions $f_A:A{\rightarrow}\{0,\;1,\;{\ldots},\;m-1\}$ and $f_B:B{\rightarrow}\{0,\;1,\;{\ldots},\;m-1\}$ such that the induced labeling on the edges $f_{E(G)}:E(G){\rightarrow}\{0,\;1,\;{\ldots},\;m-1\}$ defined by $f_{E(G)}(uv)=f_A(u)-f_B(v)$ (with respect to the ordered partition (A, B)), is also injective. In this paper we prove that odd harmonious graphs and bigraceful graphs are equivalent. We also prove that the number of distinct odd harmonious labeled graphs on m edges is m! and the number of distinct strongly odd harmonious labeled graphs on m edges is [m/2]![m/2]!. We prove that the Cartesian product of strongly odd harmonious trees is strongly odd harmonious. We find some new disconnected odd harmonious graphs.

• Honam Mathematical Journal
• /
• v.42 no.4
• /
• pp.645-651
• /
• 2020

The distinguishing number (index) D(G) (D'(G)) of a graph G is the least integer d such that G has an vertex labeling (edge labeling) with d labels that is preserved only by a trivial automorphism. The strong product G ☒ H of two graphs G and H is the graph with vertex set V (G) × V (H) and edge set {{(x1, x2),(y1, y2)}|xiyi ∈ E(Gi) or xi = yi for each 1 ≤ i ≤ 2.}. In this paper we study the distinguishing number and the distinguishing index of strong product of two graphs. We prove that for every k ≥ 2, the k-th strong power of a connected S-thin graph G has distinguishing index equal two.

• Journal of applied mathematics & informatics
• /
• v.34 no.3_4
• /
• pp.341-354
• /
• 2016

In this paper, the exact formulae for the generalized product degree distance, reciprocal product degree distance and product degree distance of tensor product of a connected graph and the complete multipartite graph with partite sets of sizes m₀, m₁, ⋯ , m_r−1 are obtained.

• Kyungpook Mathematical Journal
• /
• v.47 no.2
• /
• pp.165-714
• /
• 2007

The basis number of a graph G is the least positive integer $k$ such that G has a $k$-fold basis. In this paper, we prove that the basis number of the cartesian product of a path with a circular ladder, a M$\ddot{o}$bius ladder and path with a net is exactly 3. This improves the upper bound of the basis number of these graphs for a general theorem on the cartesian product of graphs obtained by Ali and Marougi, see [2]. Also, by this general result, the cartesian product of a theta graph with a M$\ddot{o}$bius ladder is at most 5. But in section 3 we prove that it is at most 4.

• Journal of applied mathematics & informatics
• /
• v.42 no.3
• /
• pp.625-634
• /
• 2024

The thrust of this paper is to extend the notion of Plithogenic vertex domination to the basic operations in Plithogenic product fuzzy graphs (PPFGs). When the graph is a complete PPFG, Plithogenic vertex domination numbers (PVDNs) of its Plithogenic complement and perfect Plithogenic complement are the same, since the connectivities are the same in both the graphs. Since extra edges are added to the graph in the case of perfect Plithogenic complement, the PVDN of perfect Plithogenic complement is always less than or equal to that of Plithogenic complement, when the graph under consideration is an incomplete PPFG. The maximum and minimum values of the PVDN of the intersection or the union of PPFGs depend upon the attribute values given to P-vertices, the number of attribute values and the connectivities in the corresponding PPFGs. The novelty in this study is the investigation of the variations and the relations between PVDNs in the operations of Plithogenic complement, perfect Plithogenic complement, union and intersection of PPFGs.

• Journal of the Korean Mathematical Society
• /
• v.56 no.1
• /
• pp.113-125
• /
• 2019

In this paper, we consider the Ricci curvature of a directed graph, based on Lin-Lu-Yau's definition. We give some properties of the Ricci curvature, including conditions for a directed regular graph to be Ricci-flat. Moreover, we calculate the Ricci curvature of the cartesian product of directed graphs.

• Kyungpook Mathematical Journal
• /
• v.48 no.2
• /
• pp.303-315
• /
• 2008

In [8] M. Y. Alzoubi and M. M. Jaradat studied the basis number of the composition of paths and cycles with Ladders, Circular ladders and M$\"{o}$bius ladders. Namely, they proved that the basis number of these graphs is 4 except possibly for some cases in each of them. Since the lexicographic product is noncommutative, in this paper we investigate the basis number of the lexicographic product of the different kinds of ladders with paths and cycles. In fact, we prove that the basis number of almost all of these graphs is 4.