We introduce and study the notion of a logarithmic vertex algebra, which is algebra with singularities in operator product expansion quantum fields; thus providing rigorous formulation algebraic properties fields conformal field theory. develop framework that allows many results about algebras to be extended algebras, including particular Borcherds identity Kac Existence Theorem. Several exampl...

In this paper, the structure of cocommutative vertex bialgebras is investigated. For a general bialgebra V, it proved that set G(V) group-like elements naturally an abelian semigroup, whereas P(V) primitive Lie algebra. g∈G(V), denote by Vg connected component containing g. Among main results, if V bialgebra, then V=⊕g∈G(V)Vg, where V1 subbialgebra which isomorphic to VP(V) associated algebra P...

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$, a vertex labeling $f : V(G)rightarrow mathbb{Z}_2$ induces an edge labeling $ f^{+} : E(G)rightarrow mathbb{Z}_2$ defined by $f^{+}(xy) = f(x) + f(y)$, for each edge $ xyin E(G)$.  For each $i in mathbb{Z}_2$, let $ v_{f}(i)=|{u in V(G) : f(u) = i}|$ and $e_{f^+}(i)=|{xyin E(G) : f^{+}(xy) = i}|$. A vertex labeling $f$ of a graph $G...

We continue the study of the following general problem on vertex colorings of graphs. Suppose that some vertices of a graph G are assigned to some colors. Can this “precoloring” be extended to a proper coloring of G with at most k colors (for some given k)? Here we investigate the complexity status of precoloring extendibility on some graph classes which are related to bipartite graphs, giving ...

We continue the study of the following general problem on vertex col-orings of graphs. Suppose that some vertices of a graph G are assigned to some colors. Can this \precoloring" be extended to a proper coloring of G with at most k colors (for some given k)? Here we investigate the complexity status of precoloring extendibility on some graph classes which are related to bipartite graphs, giving...

a proper vertex coloring of a simple graph is $k$-forested if the graph induced by the vertices of any two color classes is a forest with maximum degree less than $k$. a graph is $k$-forested $q$-choosable if for a given list of $q$ colors associated with each vertex $v$, there exists a $k$-forested coloring of $g$ such that each vertex receives a color from its own list. in this paper, we prov...

Proof Since PATH is co-NL-complete, it suffices to prove that PATH ∈ NL. We’ll use the the read-once certificate definition of NSPACE(·). Let Ci def = {vertices reachable from s with ≤ i steps}. Clearly, C0 = {s}. To prove “vertex x ∈ V can be reached in at most i steps starting from s” (x ∈ Ci), it’s easy by providing the path as a certificate: s, v1, · · · , v`, x where ` < i. The verifier si...

We continue the study of the following general problem on vertex col-orings of graphs. Suppose that some vertices of a graph G are assigned to some colors. Can this \precoloring" be extended to a proper coloring of G with at most k colors (for some given k)? Here we investigate the complexity status of precoloring extendibility on some graph classes which are related to bipartite graphs, giving...

Relative centricity RC values of vertices/atoms are calculated within the Distance Detour and Cluj-Distance criteria on their corresponding Shell transforms. The vertex RC distribution in a molecular graph gives atom equivalence classes, useful in interpretation of NMR spectra. Timed by vertex valences, RC provides a new index, called Centric Connectivity CC, which can be useful in the topologi...