the total version of geometric–arithmetic index of graphs is introduced based on the endvertexdegrees of edges of their total graphs. in this paper, beside of computing the total gaindex for some graphs, its some properties especially lower and upper bounds are obtained.

the zagreb indices are the oldest graph invariants used in mathematical chemistry to predictthe chemical phenomena. in this paper we define the multiple versions of zagreb indicesbased on degrees of vertices in a given graph and then we compute the first and secondextremal graphs for them.

In the extension of irregularity indices, Abdo et. al. [1] defined the total irregu-larity of a graph G = (V, E) as irrt(G) = 21 Pu,v∈V (G) du − dv, where du denotesthe vertex degree of a vertex u ∈ V (G). In this paper, we investigate the totalirregularity of trees with bounded maximal degree Δ and state integer linear pro-gramming problem which gives standard information about extremal trees a...

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...

a recently published paper [t. došlić, this journal 3 (2012) 25-34] considers the zagrebindices of benzenoid systems, and points out their low discriminativity. we show thatanalogous results hold for a variety of vertex-degree-based molecular structure descriptorsthat are being studied in contemporary mathematical chemistry. we also show that theseresults are straightforwardly obtained by using...

the rings considered in this article are  commutative  with identity which admit at least two  nonzero annihilating ideals. let $r$ be a ring. let $mathbb{a}(r)$ denote the set of all annihilating ideals of $r$ and let $mathbb{a}(r)^{*} = mathbb{a}(r)backslash {(0)}$. the annihilating-ideal graph of $r$, denoted by $mathbb{ag}(r)$  is an undirected simple graph whose vertex set is $mathbb{a}(r)...

the edge tenacity te(g) of a graph g is dened as:te(g) = min {[|x|+τ(g-x)]/[ω(g-x)-1]|x ⊆ e(g) and ω(g-x) > 1} where the minimum is taken over every edge-cutset x that separates g into ω(g - x) components, and by τ(g - x) we denote the order of a largest component of g. the objective of this paper is to determine this quantity for split graphs. let g = (z; i; e) be a noncomplete connected spli...

A set $S subseteq V(G)$ is a semitotal dominating set of a graph $G$ if it is a dominating set of $G$ andevery vertex in $S$ is within distance 2 of another vertex of $S$. Thesemitotal domination number $gamma_{t2}(G)$ is the minimumcardinality of a semitotal dominating set of $G$.We show that the semitotal domination problem isAPX-complete for bounded-degree graphs, and the semitotal dominatio...