A Roman dominating function (RDF) on a graph G is a function f : V (G) → {0, 1, 2} satisfying the condition that every vertex v for which f(v) = 0, is adjacent to at least one vertex u for which f(u) = 2. The weight of a Roman dominating function f is the value f(V (G)) = ∑ v∈V (G) f(v). The Roman domination number of G, denoted by γR(G), is the minimum weight of an RDF on G. For a given graph,...

We compute the distance-dependent two-point function of vertex-bicolored planar maps, i.e., maps whose vertices are colored in black and white so that no adjacent vertices have the same color. By distance-dependent two-point function, we mean the generating function of these maps with both a marked oriented edge and a marked vertex which are at a prescribed distance from each other. As customar...

We prove an asymptotic result on the maximum number of k-vertex subtrees in binary trees given order. This problem turns out to be equivalent determine k+2-cycles n-vertex outerplanar graphs, thus we settle generalised Turán for all cycles. also exponential growth paths Pk as a function k which implies order magnitude arbitrary trees. The bounds are strongly related sequence Catalan numbers.

A vertex irregular total k-labeling of a graph G with vertex set V and edge set E is an assignment of positive integer labels {1, 2, ..., k} to both vertices and edges so that the weights calculated at vertices are distinct. The total vertex irregularity strength of G, denoted by tvs(G)is the minimum value of the largest label k over all such irregular assignment. In this paper, we study the to...

The vertex arboricity $rho(G)$ of a graph $G$ is the minimum number of subsets into which the vertex set $V(G)$ can be partitioned so that each subset induces an acyclic graph‎. ‎A graph $G$ is called list vertex $k$-arborable if for any set $L(v)$ of cardinality at least $k$ at each vertex $v$ of $G$‎, ‎one can choose a color for each $v$ from its list $L(v)$ so that the subgraph induced by ev...

We reformulate, using super worldline formalism, the pinched gluon vertex operator proposed by Strassler. The pinched vertex operator turns out to be the product of two gluon vertex operators with the insertion of δ-function which makes the super distances between them zero. Thus the pinch procedures turn out to be nothing but the insertions of δ-function. Applying our formulation to two-loop d...

We describe two BQP-complete problems concerning properties of sparse graphs having a certain symmetry. The graphs are specified by efficiently computable functions which output the adjacent vertices for each vertex. Let i and j be two given vertices. The first problem consists in estimating the difference between the number of paths of length m from j to j and those which from i to j, where m ...

Let G be a nontrivial connected and vertex-colored graph. A subset X of the vertex set is called rainbow if any two vertices in have distinct colors. The graph vertex-disconnected for x y G, there exists S such that when are nonadjacent, belong to different components $$G-S$$ ; whereas adjacent, $$S+x$$ or $$S+y$$ $$(G-xy)-S$$ . Such an x–y vertex-cut G. For vertex-disconnection number denoted ...

A distribution function F(x) is said to be unimodal [l, p. 157] if there exists at least one value x = a such that F(x) is convex for xa. The point x = a is called the vertex of the distribution. In particular, if F(x) is absolutely continuous, then the corresponding probability density function p(x) = F'{x) is nondecreasing for xa. In the present...

Given a function f : N → R, call an n-vertex graph f-connected if separating off k vertices requires the deletion of at least f(k) vertices whenever k ≤ (n− f(k))/2. This is a common generalization of vertex connectivity (when f is constant) and expansion (when f is linear). We show that an f -connected graph contains a cycle of length linear in n if f is any linear function, contains a 1-facto...