On distance matrices of wheel graphs with odd number of vertices

2020/06/05 by Balaji, R., Bapat, R. B., Goel, Shivani
#05C50 #Combinatorics (math.CO) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2006.03289

Abstract

Let W_n denote the wheel graph having n-vertices. If i and j are any two vertices of W_n, define d_ij:= \begincases 0 amp; if~i=j
1 amp; if~i~ and ~j~ are adjacent
2 amp; else. \endcases Let D be the n × n matrix with (i,j)^{\rm th} entry equal to d_ij. The matrix D is called the distance matrix of W_n. Suppose n ≥ 5 is an odd integer. In this paper, we deduce a formula to compute the Moore-Penrose inverse of D. More precisely, we obtain an n× n matrix \widetildeL and a rank one matrix ww' such that D^† = -(1)/(2) \widetildeL+(4)/(n-1)ww'. Here, \widetildeL is positive semidefinite, \rm rank(\widetildeL)=n-2 and all row sums are equal to zero.

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