The reformulated Zagreb indices of a graph are obtained from the classical Zagreb indices by replacing vertex degrees with edge degrees, where the degree of an edge is taken as the sum of degrees of the end vertices of the edge minus 2. In this paper, we study the behavior of the reformulated first Zagreb index and apply our results to different chemically interesting molecular graphs and nano-...

The revised edge-Szeged index of a connected graph $G$ is defined as Sze*(G)=∑e=uv∊E(G)( (mu(e|G)+(m0(e|G)/2)(mv(e|G)+(m0(e|G)/2) ), where mu(e|G), mv(e|G) and m0(e|G) are, respectively, the number of edges of G lying closer to vertex u than to vertex v, the number of ed...

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...

Let S div ( n ) $S_{\rm div}(n)$ denote the set of permutations π such that for each 1 ⩽ j $1\leqslant \leqslant n$ either ∣ $j \mid \pi (j)$ or $\pi (j) j$ . These can also be viewed as vertex-disjoint directed cycle covers divisor graph D [ , ] $\mathcal {D}_{[1,n]}$ on vertices v … $v_1, \ldots v_n$ with an edge between i $v_i$ and $v_j$ if $i\mid i$ We improve recent results Pomerance by sh...

For a graph $G = (V, E)$, a partition $pi = {V_1,$ $V_2,$ $ldots,$ $V_k}$ of the vertex set $V$ is an textit{upper domatic partition} if $V_i$ dominates $V_j$ or $V_j$ dominates $V_i$ or both for every $V_i, V_j in pi$, whenever $i neq j$. The textit{upper domatic number} $D(G)$ is the maximum order of an upper domatic partition. We study the properties of upper domatic number and propose an up...

the electric quadrupole, hexadecapole moments, energy (kjmol -1) for some armchair polyhex carbon nanotubes tuvc6[2p,q] are performed by beck-lee-yang-parr [b3lyp] on 3-21g basis set using the standard procedure indices gaussian 98, then the padmakar-ivan (pi) index of tuvc6[2p,q] nanotubes in the terms of their circumference (2p) and lengh (q) is calculated and the relationships between the pa...

The Padmakar-Ivan (PI) index is a Wiener-Szeged-like topological index which reflects certain structural features of organic molecules. The PI index of a graph G is the sum of all edges uv of G of the number of edges which are not equidistant from the vertices u and v. In this paper we obtain the second and third extremals of catacondensed hexagonal systems with respect to the PI index.

The electric quadrupole, hexadecapole moments, energy (kJmol -1) for some armchair polyhex carbon nanotubes TUVC6[2p,q] are performed by Beck-Lee-Yang-Parr [B3LYP] on 3-21G basis set using the standard procedure indices GAUSSIAN 98, then the Padmakar-Ivan (PI) index of TUVC6[2p,q] nanotubes in the terms of their circumference (2p) and lengh (q) is calculated and the relationships between the Pa...

Food prices are expected to affect dietary intakes, however, previous findings are mixed and few are based on nationally representative data. We examined the associations of price indices of fast foods (FF-PI) and fruits and vegetables (FV-PI) with dietary intakes and BMI among U.S. children and adolescents using data from the Continuing Survey of Food Intakes by Individuals (CSFII; 1994-1998) ...

The edge Szeged index of a connected graph G is defined as the sum of products mu(e|G)mv(e|G) over all edges e = uv of G, where mu(e|G) is the number of edges whose distance to vertex u is smaller than the distance to vertex v, and mv(e|G) is the number of edges whose distance to vertex v is smaller than the distance to vertex u. In this paper, we determine the n-vertex unicyclic graphs with th...