An even (resp. odd) lollipop is the coalescence of a cycle of even (resp. odd) length and a path with pendant vertex as distinguished vertex. It is known that the odd lollipop is determined by its spectrum and the question is asked by W. Haemers, X. Liu and Y. Zhang for the even lollipop. We revisit the proof for odd lollipop, generalize it for even lollipop and therefore answer the question. O...

A group of destructive changes occurring in jaws in patients with maxillary complete dentures and mandibular removable partial dentures (bilaterally) has been described in the literature as the combination syndrome. However, this condition is not clinically observed in all patients. The aim of this study was to establish the prevalence index on signs of combination syndrome and to verify whethe...

A wheel is a graph consisting of an induced cycle length at least four and single additional vertex with three neighbours on the cycle. We prove that no Burling contains wheel. graphs are triangle-free have arbitrarily large chromatic number, so this answers question Trotignon disproves conjecture Scott Seymour.

The well-known theorem of Erdős and Pósa says that a graph G has either k vertex-disjoint cycles or a vertex set X of order at most f(k) such that G\X is a forest. Starting with this result, there are many results concerning packing and covering cycles in graph theory and combinatorial optimization. In this paper, we generalize Erdős-Pósa’s result to cycles that are required to go through a set...

Let k be a positive integer and G be a k-connected graph. In 2009, Chartrand, Johns, McKeon, and Zhang introduced the rainbow k-connection number rck(G) of G. An edge-coloured path is rainbow if its edges have distinct colours. Then, rck(G) is the minimum number of colours required to colour the edges of G so that any two vertices of G are connected by k internally vertex-disjoint rainbow paths...

1 DDS, Resident Fellow in Removable Partial Dentures, Faculty of Dental Medicine, University of Monastir, Tunisia. 2 DDS, Professor in Histology and Embryology, Faculty of Dental Medicine, University of Monastir, Tunisia. 3 DDS, Assistant Professor in Removable Partial Dentures, Faculty of Dental Medicine, University of Monastir, Tunisia. 4 DDS, Professor in Removable Partial Dentures, Faculty ...

The complete graph on n vertices is denoted by K, and the graph on n vertices in which each pair of vertices is joined by exactly 2 edges is denoted by 2K,. An m-cycle is a sequence of m distinct vertices (u1, u2, . . . . u,) such that ui is adjacent to ui+ 1 and U, is adjacent to u1 . A spanning subgraph H of G is a subgraph for which V(H) = V(G). An i-factor of a graph G is a spanning subgrap...

A graph G of order n is called arbitrarily vertex decomposable if for each sequence (n1, . . . , nk) of positive integers such that ∑k i=1 ni = n, there exists a partition (V1, . . . , Vk) of vertex set of G such that for every i ∈ {1, . . . , k} the set Vi induces a connected subgraph of G on ni vertices. We consider arbitrarily vertex decomposable unicyclic graphs with dominating cycle. We al...

Let P be a graph property. A graph G is said to be locally P (closed locally P) if the subgraph induced by the open neighbourhood (closed neighbourhood, respectively) of every vertex in G has property P. The clustering coefficient of a vertex is the proportion of pairs of its neighbours that are themselves neighbours. The minimum clustering coefficient of G is the smallest clustering coefficien...