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A graph associated to proper non-small ideals of a commutative ring

S. Ebrahimi Atani, S. Dolati Pish Hesari, M. Khoramdel (2017)

Commentationes Mathematicae Universitatis Carolinae

In this paper, a new kind of graph on a commutative ring is introduced and investigated. Small intersection graph of a ring R , denoted by G ( R ) , is a graph with all non-small proper ideals of R as vertices and two distinct vertices I and J are adjacent if and only if I ∩ J is not small in R . In this article, some interrelation between the graph theoretic properties of this graph and some algebraic properties of rings are studied. We investigated the basic properties of the small intersection graph as diameter,...

A lower bound for the 3-pendant tree-connectivity of lexicographic product graphs

Yaping Mao, Christopher Melekian, Eddie Cheng (2023)

Czechoslovak Mathematical Journal

For a connected graph G = ( V , E ) and a set S ⊆ V ( G ) with at least two vertices, an S -Steiner tree is a subgraph T = ( V ' , E ' ) of G that is a tree with S ⊆ V ' . If the degree of each vertex of S in T is equal to 1, then T is called a pendant S -Steiner tree. Two S -Steiner trees are internally disjoint if they share no vertices other than S and have no edges in common. For S ⊆ V ( G ) and | S | ≥ 2 , the pendant tree-connectivity τ G ( S ) is the maximum number of internally disjoint pendant S -Steiner trees in G , and for k ≥ 2 , the k -pendant tree-connectivity τ k ( G ) ...

A New Method for Computing the Eccentric Connectivity Index of Fullerenes

Ghorbani, Modjtaba, Malekjani, Khadijeh (2012)

Serdica Journal of Computing

ACM Computing Classification System (1998): G.2.2, G.2.3.The eccentric connectivity index of the molecular graph G, ξ^c (G), was proposed by Sharma, Goswami and Madan. It is defined as ξ^c (G) = Σu∈V(G)degG(u) ecc(u), where degG(x) denotes the degree of the vertex x in G and ecc(u) = Max{d(x, u) | x ∈ V (G)}. In this paper this graph invariant is computed for an infinite class of fullerenes by means of group action.

A New Proof that 4-Connected Planar Graphs are Hamiltonian-Connected

Xiaoyun Lu, Douglas B. West (2016)

Discussiones Mathematicae Graph Theory

We prove a theorem guaranteeing special paths of faces in 2-connected plane graphs. As a corollary, we obtain a new proof of Thomassen’s theorem that every 4-connected planar graph is Hamiltonian-connected.

A Note on Barnette’s Conjecture

Jochen Harant (2013)

Discussiones Mathematicae Graph Theory

Barnette conjectured that each planar, bipartite, cubic, and 3-connected graph is hamiltonian. We prove that this conjecture is equivalent to the statement that there is a constant c > 0 such that each graph G of this class contains a path on at least c|V (G)| vertices.

A note on face coloring entire weightings of plane graphs

Stanislav Jendrol, Peter Šugerek (2014)

Discussiones Mathematicae Graph Theory

Given a weighting of all elements of a 2-connected plane graph G = (V,E, F), let f(α) denote the sum of the weights of the edges and vertices incident with the face _ and also the weight of _. Such an entire weighting is a proper face colouring provided that f(α) ≠ f(β) for every two faces α and _ sharing an edge. We show that for every 2-connected plane graph there is a proper face-colouring entire weighting with weights 1 through 4. For some families we improved 4 to 3

A proof of menger's theorem by contraction

Frank Göring (2002)

Discussiones Mathematicae Graph Theory

A short proof of the classical theorem of Menger concerning the number of disjoint AB-paths of a finite graph for two subsets A and B of its vertex set is given. The main idea of the proof is to contract an edge of the graph.

A simple proof of Whitney's Theorem on connectivity in graphs

Kewen Zhao (2011)

Mathematica Bohemica

In 1932 Whitney showed that a graph G with order n ≥ 3 is 2-connected if and only if any two vertices of G are connected by at least two internally-disjoint paths. The above result and its proof have been used in some Graph Theory books, such as in Bondy and Murty’s well-known Graph Theory with Applications. In this note we give a much simple proof of Whitney’s Theorem.

A Triple of Heavy Subgraphs Ensuring Pancyclicity of 2-Connected Graphs

Wojciech Wide (2017)

Discussiones Mathematicae Graph Theory

A graph G on n vertices is said to be pancyclic if it contains cycles of all lengths k for k ∈ {3, . . . , n}. A vertex v ∈ V (G) is called super-heavy if the number of its neighbours in G is at least (n+1)/2. For a given graph H we say that G is H-f1-heavy if for every induced subgraph K of G isomorphic to H and every two vertices u, v ∈ V (K), dK(u, v) = 2 implies that at least one of them is super-heavy. For a family of graphs H we say that G is H-f1-heavy, if G is H-f1-heavy for every graph...

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