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# Article

 Title: On the Boolean function graph of a graph and on its complement (English) Author: Janakiraman, T. N. Author: Muthammai, S. Author: Bhanumathi, M. Language: English Journal: Mathematica Bohemica ISSN: 0862-7959 (print) ISSN: 2464-7136 (online) Volume: 130 Issue: 2 Year: 2005 Pages: 113-134 Summary lang: English . Category: math . Summary: For any graph $G$, let $V(G)$ and $E(G)$ denote the vertex set and the edge set of $G$ respectively. The Boolean function graph $B(G,L(G),\mathop {\mathrm NINC})$ of $G$ is a graph with vertex set $V(G)\cup E(G)$ and two vertices in $B(G,L(G),\mathop {\mathrm NINC})$ are adjacent if and only if they correspond to two adjacent vertices of $G$, two adjacent edges of $G$ or to a vertex and an edge not incident to it in $G$. For brevity, this graph is denoted by $B_1(G)$. In this paper, structural properties of $B_1(G)$ and its complement including traversability and eccentricity properties are studied. In addition, solutions for Boolean function graphs that are total graphs, quasi-total graphs and middle graphs are obtained. (English) Keyword: eccentricity Keyword: self-centered graph Keyword: middle graph Keyword: Boolean function graph MSC: 05C12 MSC: 05C15 MSC: 05C45 MSC: 05C75 MSC: 06E30 idZBL: Zbl 1110.05086 idMR: MR2148646 DOI: 10.21136/MB.2005.134130 . Date available: 2009-09-24T22:19:10Z Last updated: 2020-07-29 Stable URL: http://hdl.handle.net/10338.dmlcz/134130 . Reference: [1] J. Akiyama, T. Hamada, I. Yoshimura: On characterizations of the middle graphs.TRU Math. 11 (1975), 35–39. MR 0414436 Reference: [2] M. Behzad: A criterion for the planarity of the total graph of a graph.Proc. Camb. Philos. Soc. 63 (1967), 679–681. Zbl 0158.20703, MR 0211896, 10.1017/S0305004100041657 Reference: [3] J. A. Bondy, U. S. Murty: Graph Theory with Application.Macmillan, London, 1976. Reference: [4] S. B. Chikkodimath, E. Sampathkumar: Semi-total graphs II.Graph Theory Research Report, Karnatak University 2 (1973), 5–9. Reference: [5] T. Hamada, I. Yoshimura: Traversability and connectivity of the middle graph of a graph.Discrete Math. 14 (1976), 247–256. MR 0414435, 10.1016/0012-365X(76)90037-6 Reference: [6] F. Harary: Graph Theory.Addison-Wesley, Reading, Mass., 1969. Zbl 0196.27202, MR 0256911 Reference: [7] E. Sampathkumar, Prabha S. Neeralagi: The neighborhood number of a graph.Indian J. Pure Appl. Math. 16 (1985), 126–132. MR 0780299 Reference: [8] D. V. S. Sastry, B. Syam Prasad Raju: Graph equations for line graphs, total graphs, middle graphs and quasi-total graphs.Discrete Math. 48 (1984), 113–119. MR 0732207, 10.1016/0012-365X(84)90137-7 Reference: [9] H. Whitney: Congruent graphs and the connectivity of graphs.Am. J. Math. 54 (1932), 150–168. Zbl 0003.32804, MR 1506881, 10.2307/2371086 .

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