Rainbow cycles and paths in fan and wheel graph

Fitriani, R.; Sugeng, Kiki Ariyanti; Hariadi, Nora

doi:10.1063/1.5064193

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2020

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“…The aim of this paper is to obtain explicitly the QE constant of a fan graph K 1 + P n , that is the graph join of the singleton graph K 1 and the path P n with n ≥ 1. The fan graphs form a basic family of graphs and have been studied in various contexts, for example, vertex coloring [7], edge coloring [8], chromatic polynomials [16], inversion of the distance matrices [9], graph spectrum [14] and so forth. Although a fan graph has a relatively simple structure at a first glance, no reasonable formula for QEC(K 1 + P n ) has been obtained.…”

Section: Introductionmentioning

confidence: 99%

On quadratic embedding constants of star product graphs

Młotkowski

Obata

2020

Hokkaido Math. J.

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We derive a general formula for the quadratic embedding constant of a graph join Km + G, where Km is the empty graph on m ≥ 1 vertices and G is an arbitrary graph. Applying our formula to a fan graph K 1 + P n , where K 1 = K1 is the singleton graph and P n is the path on n ≥ 1 vertices, we show that QEC(K 1 + P n ) = − αn − 2, where αn is the minimal zero of a new polynomial Φ n (x) related to Chebyshev polynomials of the second kind. Moreover, for an even n we have αn = min ev(A n ), where the right-hand side is the An minimal eigenvalue of the adjacency matrix A n of P n . For an odd n we show that min ev(A n+1 ) ≤ αn < min ev(A n ).

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