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- W2057362364 abstract "The line graph of a graph G is defined to be the graph whose vertices are the edges of G , two vertices being adjacent if and only if the corresponding edges of G have a common vertex. If π is a projective plane of order n , then the bipartite graph of π , denoted by G(π) , is the graph whose vertices are the 2( n 2 +n +1) points and lines of π , two vertices of G(π) being adjacent if and only if one is a point and the other is a line containing the point. The line graph of G(π) will be denoted by L(π) . A characterization of the class { L(π) } in terms of the eigenvalues of the adjacency matrix of L(π) has been given by Hoffman. An alternative characterization of the class { L(π) } is presented here. If d(x, y) denotes the distance between two vertices x and y and Δ(x, y) denotes the number of vertices adjacent to both x and y , then G=L(π) has the following properties: (b 1 ) The number of vertices in G is (n+1)(n 2 +n+1) . (b 2 ) G is connected and regular of valence 2n . (b 3 ) Δ(x, y)=n −1 if d(x, y) =1 (b 4 ) Δ(x, y) =1 if d(x, y) =2. (b 5 ) If d(x, y) =2, the number of vertices z , such that d(x, z) =1 and d(y, z) =2, is at most n −1. In this paper we show that any graph G with no loops and no multiple edges having the properties (b 1 )−(b 5 ) must be the line graph of a projective plane of order n ." @default.
- W2057362364 created "2016-06-24" @default.
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- W2057362364 date "1967-12-01" @default.
- W2057362364 modified "2023-10-16" @default.
- W2057362364 title "A geometric characterization of the line graph of a projective plane" @default.
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- W2057362364 doi "https://doi.org/10.1016/s0021-9800(67)80106-6" @default.
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