On graphs whose Wiener complexity equals their order and on Wiener index of asymmetric graphs

Yaser Alizadeh and Sandi Klavžar

Applied Mathematics and Computation, 2018, vol. 328, issue C, 113-118

Abstract: If u is a vertex of a graph G, then the transmission of u is the sum of distances from u to all the other vertices of G. The Wiener complexity CW(G) of G is the number of different complexities of its vertices. G is transmission irregular if CW(G)=n(G). It is proved that almost no graphs are transmission irregular. Let Tn1,n2,n3 be the tree obtained from paths of respective lengths n1, n2, and n3, by identifying an end-vertex of each of them. It is proved that T1,n2,n3 is transmission irregular if and only if n3=n2+1 and n2∉{(k2−1)/2,(k2−2)/2} for some k ≥ 3. It is also proved that if T is an asymmetric tree of order n, then the Wiener index of T is bounded by (n3−13n+48)/6 with equality if and only if T=T1,2,n−4. A parallel result is deduced for asymmetric uni-cyclic graphs.

Keywords: Wiener index; Wiener complexity; Asymmetric graphs; Trees; Uni-cyclic graphs (search for similar items in EconPapers)
Date: 2018
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (12)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:328:y:2018:i:c:p:113-118

DOI: 10.1016/j.amc.2018.01.039

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