Let ∆(G), ∆ for short, be the maximum degree of a graph G. In this paper, trees (resp., unicyclic graphs and bicyclic graphs), which attain the first and the second largest spectral radius with respect to the adjacency matrix in the class of trees (resp., unicyclic graphs and bicyclic graphs) with n vertices and the maximum degree ∆, where ∆ ≥ n+1 2 (resp., ∆ ≥ n 2 +1 and ∆ ≥ n+3 2 ) are determ...

The title compound, C9H12O5, is a bicyclic lactone, presenting a 2,6-dioxabi-cyclo-[3.3.0]octan-3-one skeleton, which was obtained through an intra-molecular lactonization. The bicyclic lactone presents a cis ring-junction and a 1,5-trans-substituted tetra-hydro-furan. Both five-membered rings are in twisted envelope conformations with one of the fused C atoms as the flap. The dihedral angle be...

The nullity of a graph is the multiplicity of the eigenvalues zero in its spectrum. A signed graph is a graph with a sign attached to each of its edges. In this paper, we obtain the coefficient theorem of the characteristic polynomial of a signed graph, give two formulae on the nullity of signed graphs with cut-points. As applications of the above results, we investigate the nullity of the bicy...

It is known that regular digraphs of degree d, diameter k and unique walks of length not smaller than h and not greater than k between all pairs of vertices ([ h, k ]-digraphs), exist only for h k and h = k 1, if d ;::: 2. This paper deals with the problem of the enumeration of [k 1, kJ-digraphs in the case of diameter k = 2 or degree d = 2. It is shown, using algebraic techniques, that the lin...

A natural digraph analogue of the graph-theoretic concept of an ‘independent set’ is that of an ‘acyclic set’, namely a set of vertices not spanning a directed cycle. Hence a digraph analogue of a graph coloring is a decomposition of the vertex set into acyclic sets and we say a digraph is uniquely n-colorable when this decomposition is unique up to relabeling. It was shown probabilistically in...

Let $R$ be a commutative ring. In this paper, by using algebraic properties of $R$, we study the Hase digraph of prime ideals of $R$.

Given a digraph D = (V, A), a quasi-kernel of D is an independent set Q C_ V such that for every vertex v not contained in Q, either there exists a vertex u E Q such that v dominates u, or there exists a vertex w such that v dominates w and w dominates some vertex u E Q. A sink in a digraph D = (V, A) is a vertex v E V that dominates no vertex of D. In this thesis we prove that if D is a semico...

A digraph D is called a kernel-perfect digraph or KP -digraph when every induced subdigraph of D has a kernel. We call the digraph D an m-coloured digraph if the arcs of D are coloured with m distinct colours. A path P is monochromatic in D if all of its arcs are coloured alike in D. The closure of D, denoted by ζ(D), is the m-coloured digraph defined as follows: V (ζ(D)) = V (D), and A (ζ(D)) ...