The center (periphery) of a graph is the set of vertices with minimum (maximum) eccentricity. In this paper, the structure of centers and peripheries of some classes of composite graphs are determined. The relations between eccentricity, radius and diameter of such composite graphs are also investigated. As an application we determine the center and periphery of some chemical graphs such as nan...

A graph G is well-covered if all its maximal independent sets are of the same cardinality. Assume that a weight function w is defined on its vertices. Then G is w-well-covered if all maximal independent sets are of the same weight. For every graph G, the set of weight functions w such that G is w-well-covered is a vector space. Given an input graph G without cycles of length 4, 5, and 6, we cha...

The $k$-token graph $T_k(G)$ is the whose vertices are $k$-subsets of a $G$, with two adjacent if their symmetric difference an edge $G$. We explore when well-covered graph, that is, all its maximal independent sets have same cardinality. For bipartite graphs we classify well-covered. arbitrary show $T_2(G)$ well-covered, then girth $G$ at most four. include upper and lower bounds on independen...

A graph is well-covered if every independent set can be extended to a maximum independent set. We show that it is co-NP-complete to determine whether an arbitrary graph is well-covered, even when restricted to the family of circulant graphs. Despite the intractability of characterizing the complete set of well-covered circulant graphs, we apply the theory of independence polynomials to show tha...

If sk equals the number of stable sets of cardinality k in the graph G, then I(G; x) = α(G) ∑ k=0 skx k is the independence polynomial of G (Gutman and Harary, 1983). Alavi, Malde, Schwenk and Erdös (1987) conjectured that I(G; x) is unimodal whenever G is a forest, while Brown, Dilcher and Nowakowski (2000) conjectured that I(G; x) is unimodal for any well-covered graph G. Michael and Traves (...

Little [12] showed that, in a certain sense, the only minimal non-Pfaffian bipartite matching covered graph is the brace K3,3. Using a stronger notion of minimality than the one used by Little, we show that every minimal nonPfaffian brick G contains two disjoint odd cycles C1 and C2 such that the subgraph G − V (C1 ∪ C2) has a perfect matching. This implies that the only minimal non-Pfaffian so...

‎The excessive index of a bridgeless cubic graph $G$ is the least integer $k$‎, ‎such that $G$ can be covered by $k$ perfect matchings‎. ‎An equivalent form of Fulkerson conjecture (due to Berge) is that every bridgeless‎ ‎cubic graph has excessive index at most five‎. ‎Clearly‎, ‎Petersen graph is a cyclically 4-edge-connected snark with excessive index at least 5‎, ‎so Fouquet and Vanherpe as...

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In 1987, Lovv asz conjectured that every brick G diierent from K 4 , C 6 , and the Petersen graph has an edge e such that G e is a matching covered graph with exactly one brick. Lovv asz and Vempala announced a proof of this conjecture in 1994. Their paper is under preparation. We present here an independent proof of their theorem. We shall in fact prove that if G is any brick diierent from K 4...