THE STRUCTURE OF GRAPHS ON n VERTICES WITH THE DEGREE SUM OF ANY TWO NONADJACENT VERTICES EQUAL TO n-2
Keyword(s):
Let G be an undirected simple graph on n vertices and sigma2(G)=n-2 (degree sum of any two non-adjacent vertices in G is equal to n-2) and alpha(G) be the cardinality of an maximum independent set of G. In G, a vertex of degree (n-1) is called total vertex. We show that, for n>=3 is an odd number then alpha(G)=2 and G is a disconnected graph; for n>=4 is an even number then 2=<alpha(G)<=(n+2)/2, where if alpha(G)=2 then G is a disconnected graph, otherwise G is a connected graph, G contains k total vertices and n-k vertices of degree delta=(n-2)/2, where 0<=k<=(n-2)/2. In particular, when k=0 then G is an (n-2)/2-Regular graph.
1997 ◽
Vol 48
(6)
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pp. 612-622
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2013 ◽
Vol 30
(4)
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pp. 1173-1179
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2014 ◽
Vol 56
(1)
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pp. 197-219
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