## Abstract Let __G__ be a simple graph of order __n__ with Laplacian spectrum {Ξ»~__n__~, Ξ»~__n__β1~, β¦, Ξ»~1~} where 0=Ξ»~__n__~β€Ξ»~__n__β1~β€β β€Ξ»~1~. If there exists a graph whose Laplacian spectrum is __S__={0, 1, β¦, __n__β1}, then we say that __S__ is Laplacian realizable. In 6, Fallat et al. posed
Harary's conjectures on integral sum graphs
β Scribed by Zhibo Chen
- Publisher
- Elsevier Science
- Year
- 1996
- Tongue
- English
- Weight
- 167 KB
- Volume
- 160
- Category
- Article
- ISSN
- 0012-365X
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π SIMILAR VOLUMES
Our purpose is to consider the following conjectures: Conjecture 1 (Barneffe). . Every cubic 3-connected bipartite planar graph is Hamiltonian. Conjecture 2 (Jaeger). Every cubic cyclically 4-edge connected graph G has a cycle C such that G -V(C) is acyclic. Conjecture 3 (Jackson, Fleischner). Ever
## Abstract Rao posed the following conjecture, βLet G be a selfβcomplementary graph of order __p__, Ο = (d~1~ β¦ dp) be its degree sequence. Then G has a kβfactor if and only if Ο β k, = (d1 β k, β¦ dP β k) is graphical.β We construct a family of counterexamples for this conjecture for every k β©Ύ 3.
Let F be a connected graph. F is said to be interval-regular if I F~\_ l(u) uF(x )J =. i holds for all vertices u and x ~ Fi(u), i > 0. For u, v e F, let I (u, v) denote the set of all vertices on a shortest path connecting u, v. A subset W of V(F) is said to be convex if l(u,v) c W holds for each u
## Abstract Faudree and Schelp conjectured that for any two vertices __x, y__ in a Hamiltonianβconnected graph __G__ and for any integer __k__, where __n__/2 β©½ __k__ β©½ __n__ β 1, __G__ has a path of length __k__ connecting __x__ and __y__. However, we show in this paper that there are infinitely ma