کد مقاله کد نشریه سال انتشار مقاله انگلیسی نسخه تمام متن
5777019 1413649 2017 14 صفحه PDF دانلود رایگان
عنوان انگلیسی مقاله ISI
Proofs of three conjectures on the quotients of the (revised) Szeged index and the Wiener index and beyond
موضوعات مرتبط
مهندسی و علوم پایه ریاضیات ریاضیات گسسته و ترکیبات
پیش نمایش صفحه اول مقاله
Proofs of three conjectures on the quotients of the (revised) Szeged index and the Wiener index and beyond
چکیده انگلیسی
Let W(G),Sz(G) and Sz∗(G) be the Wiener index, Szeged index and revised Szeged index of a connected graph G, respectively. Call Ln,r a lollipop if it is obtained by identifying a vertex of Cr with an end-vertex of Pn−r+1. For a connected unicyclic graph G with n≥4 vertices, Hansen et al. (2010) conjectured: (A)Sz(G)W(G)≤2−8n2+7,if n is odd,2,if n is even,(B)Sz∗(G)W(G)≥1+3(n2+4n−6)2(n3−7n+12),if n≤9,1+24(n−2)n3−13n+36,if n≥10,(C)Sz∗(G)W(G)≤2+2n2−1,if n is odd,2,if n is even,where the equality in (A) holds if and only if G is the lollipop Ln,n−1 if n is odd, and the cycle Cn if n is even; the equality in (B) holds if and only if G is the lollipop Ln,3 if n≤9, and Ln,4 if n≥10, whereas the equality in (C) holds if and only if G is the cycle Cn. In this paper, we not only confirm these conjectures but also determine the lower bound of Sz∗(G)∕W(G) (resp. Sz(G)∕W(G)) for cyclic graphs G. The extremal graphs that achieve these lower bounds are characterized.
ناشر
Database: Elsevier - ScienceDirect (ساینس دایرکت)
Journal: Discrete Mathematics - Volume 340, Issue 3, March 2017, Pages 311-324
نویسندگان
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