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The neighbour-integrity of total graphs
International Journal of Computer Mathematics, 2000Let Gbe a graph. A vertex subversion strategy of G, say S, is a set of vertices in G whose closed neighbourhood is removed from G. The survival-subgraph is denoted by G/S. The neighbour-integrity of G NI(G), is defined to be , where S is any vertex subversion strategy of G, and c(G/S) is the maximum order of the components of G/S.
Kirlangic A., Ozan A.
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Uniquely Total Colorable Graphs
Graphs and Combinatorics, 1997A total coloring of a graph is an assignment of colors to the vertices and edges of the graph so that no two adjacent edges have the same color, no two adjacent vertices have the same color and no vertex and an incident edge have the same color. The minimum number of colors needed by a total coloring is called the total chromatic number and is denoted \
Saieed Akbari +3 more
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On the toughness of the total graph of a graph.
Ars Comb., 2001Let \(G\) be a graph. If \(G\) is not a complete graph, then its toughness \(t(G)\) is defined by the formula \(t(G) = \min \{| S| / \omega (G - S) \mid S \subset V (G),\,\omega (G - S) \geq 2\}\), where \(\omega (G - S)\) is the number of connected components of \(G - S\). If \(G\) is a complete graph, then \(t(G)\) is defined as \(+ \infty \).
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Ars Comb., 1996
A total dominating set in a graph \(G\) is a subset \(D\) of the vertex set \(V(G)\) of \(G\) with the property that for each vertex \(x\in V(G)\) there exists a vertex \(y\in D\) adjacent to \(x.\) The minimum number of vertices of a total dominating set in \(G\) is the total domination number \(\gamma _t(G)\) of \(G.\) The symbol \(\overline G ...
S. Arumugam, A. Thuraiswamy
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A total dominating set in a graph \(G\) is a subset \(D\) of the vertex set \(V(G)\) of \(G\) with the property that for each vertex \(x\in V(G)\) there exists a vertex \(y\in D\) adjacent to \(x.\) The minimum number of vertices of a total dominating set in \(G\) is the total domination number \(\gamma _t(G)\) of \(G.\) The symbol \(\overline G ...
S. Arumugam, A. Thuraiswamy
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Total Colorings of Product Graphs
Graphs and Combinatorics, 2018A proper total coloring of a graph \(G\) is an assignment of colors to vertices and edges of the graph, such that adjacent and incident elements receive different colors. The total chromatic number \(\chi^{\prime\prime}(G)\) of the graph \(G\) is the minimum number of colors needed for a proper total coloring.
J. Geetha 0001, K. Somasundaram 0001
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A class of totally antimagic total graphs [PDF]
Australas. J Comb., 2016 Summary: A total labeling of a graph \(G\) is a bijection from the vertex set and edge set of \(G\) onto the set \(\{1,2,\dots,|V(G)|+|E(G)|\}\). Such a labeling \(\xi\) is vertex-antimagic (edge-antimagic) if all vertex-weights \(wt\xi (v)=\xi(v)+\sum_{vu\in E(G)}\xi(vu)\), \(v\in V(G)\), (all edge-weights \(wt_\xi(vu)=\xi(v)+\xi(vu)+\xi(u)\), \(vu\in
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A Comprehensive Survey on Graph Neural Networks
IEEE Transactions on Neural Networks and Learning Systems, 2021Chengqi Zhang, Philip Yu, Shirui Pan
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