2020/12/08 by Das, Joyentanuj, Mohanty, Sumit · 1 citation
#05C12 #05C50 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2012.04341
Let G = Kn1,n2,⋯,nt be a complete t-partite graph on n=∑i=1t ni vertices. The distance between vertices i and j in G, denoted by dij is defined to be the length of the shortest path between i and j. The squared distance matrix Δ(G) of G is the n× n matrix with (i,j)th entry equal to 0 if i = j and equal to dij2 if i ≠ j. We define the squared distance energy EΔ(G) of G to be the sum of the absolute values of its eigenvalues. We determine the inertia of Δ(G) and compute the squared distance energy EΔ(G). More precisely, we prove that if ni ≥ 2 for 1≤ i ≤ t, then EΔ(G)=8(n-t) and if h= |\i : ni=1\|≥ 1, then 8(n-t)+2(h-1) ≤ EΔ(G) lt; 8(n-t)+2h. Furthermore, we show that for a fixed value of n and t, both the spectral radius of the squared distance matrix and the squared distance energy of complete t-partite graphs on n vertices are maximal for complete split graph Sn,t and minimal for Turán graph Tn,t.