# On the Neighbour Sum Distinguishing Index of Graphs with Bounded Maximum Average Degree

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A signed graph (or, in short, sigraph) S = (Su, s) consists of an underlying graph Su := G = (V, E) and a function Ïƒ : E(Su) â†’ {+, -}, called the signature of S. A marking of S is a function Î¼: V(S) â†’ {+, -}. The canonical marking of a signed graph S, denoted mÎ¼;Ïƒ, is...

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The n-dimensional hypercube Qn is bipancyclic; that is, it contains a cycle of every even length from 4 to 2n. In this paper, we prove that Qnâ€‰â€‰(nâ‰¥3) contains a 3-regular, 3-connected, bipancyclic subgraph with l vertices for every even l from 8 to 2n except 10.

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A subset D of V is called an equitable dominating set [8] if for every v âˆˆ V -D there exists a vertex u âˆˆ D such that uv âˆˆ E(G) and |deg(u) - deg(v)| â‰¤ 1, where deg(u) denotes the degree of vertex u and deg(v) denotes the degree of vertex v. Recently, The minimum covering...

- Further Properties of Trees with Minimal Atom-Bond Connectivity Index. Jianping Liu; Jinsong Chen // Abstract & Applied Analysis;2014, p1
Let G = (V, E) be a graph the atom-bond connectivity (ABC) index is defined as the sum of weights ((du + dv - 2 ) /dudv)1/2 over all edges uv of G, where du denotes the degree of a vertex u of G. In this paper, we determined a few structural features of the trees with minimal ABC index also we...