• Kyungpook Mathematical Journal
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• v.43 no.4
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• pp.567-577
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• 2003

In this paper, we classify the Steinhaus graphs with minimum degree two.

• Journal of applied mathematics & informatics
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• v.30 no.5_6
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• pp.793-807
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• 2012

If G is any connected graph of order p; then the thorn graph $G_p^*$ with code ($n_1$, $n_2$, ${\cdots}$, $n_p$) is obtained by adding $n_i$ pendent vertices to each $i^{th}$ vertex of G. By treating the pendent edge of a thorn graph as $P_2$, $K_2$, $K_{1,1}$, $K_1{\circ}K_1$ or $P_1{\circ}K_1$, we generalize a thorn graph by replacing $P_2$ by $P_m$, $K_2$ by $K_m$, $K_{1,1}$ by $K_{m,n}$, $K_1{\circ}K_1$ by $K_m{\circ}K_1$ and $P_1{\circ}K_1$ by $P_m{\circ}K_1$ and their respective generalized thorn graphs are denoted by $G_P$, $G_K$, $G_B$, $G_{KK}$ and $G_{PK}$ respectively. Many chemical compounds can be treated as $G_P$, $G_K$, $G_B$, $G_{KK}$ and $G_{PK}$ of some graphs in graph theory. In this paper, we obtain the bounds of the wiener index for these generalization of thorn graphs.

• Journal of applied mathematics & informatics
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• v.41 no.4
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• pp.839-848
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• 2023

Let G be a simple undirected graph. A planar graph known as a Halin graph(HG) is characterised by having three connected and pendent vertices of a tree that are connected by an outer cycle. A subset S of V is said to be a dominating set of the graph G if each vertex u that is part of V is dominated by at least one element v that is a part of S. The domination number of a graph is denoted by the γ(G), and it corresponds to the minimum size of a dominating set. A dominating set S is called a secure dominating set if for each v ∈ V＼S there exists u ∈ S such that v is adjacent to u and S₁ = (S＼{v}) ∪ {u} is a dominating set. The minimum cardinality of a secure dominating set of G is equal to the secure domination number γ_s(G). In this article we found the secure domination number of Halin graph(HG) with perfet k-ary tree and also we determined secure domination of rooted product of special trees.