Let $G$ be a graph and let $m_{i,j}(G)$, $i,jge 1$, be the number of edges $uv$ of $G$ such that ${d_v(G), d_u(G)} = {i,j}$. The $M$-polynomial of $G$ is $M(G;x,y) = sum_{ile j} m_{i,j}(G)x^iy^j$. With $M(G;x,y)$ in hands, numerous degree-based topological indices of $G$ can be routinely computed. In this note a formula for the $M$-polynomial of planar (chemical) graphs which have only vertices...

the wiener index w(g) of a connected graph g is defined as the sum of the distances betweenall unordered pairs of vertices of g. the eccentricity of a vertex v in g is the distance to avertex farthest from v. in this paper we obtain the wiener index of a graph in terms ofeccentricities. further we extend these results to the self-centered graphs.

Let G = L ∪ R be a bipartite graph with n left vertices and m right vertices. Then G corresponds to a binary code CG with block length n, and dimension k ≥ n −m, where the left vertices correspond to variables x1, . . . , xn ∈ {0, 1}, while the right vertices c1, . . . , cm correspond to constraints, such that ci is satisfied iff ∑ u↔ci xu = 0. Let SAT be the set of satisfied constraints, and U...

let g = (v, e) be a simple graph. hosoya polynomial of g isd(u,v)h(g, x) = {u,v}v(g)x , where, d(u ,v) denotes the distance between vertices uand v. as is the case with other graph polynomials, such as chromatic, independence anddomination polynomial, it is natural to study the roots of hosoya polynomial of a graph. inthis paper we study the roots of hosoya polynomials of some specific graphs.

We consider exact distance oracles for directed weighted planar graphs in the presence of failing vertices. Given a source vertex u , target v and set X k failed vertices, such an oracle returns length shortest -to- path that avoids all vertices . propose can handle any number failures. show several tradeoffs between space, query time, preprocessing time. In particular, graph with n constant we...

Let G be a connected graph with n vertices. Then the class of graphs having vertices is denoted by Gn. The subclass 5 cycles are Gn5. classification G∈Gn5 depends on number edges and sum degrees graph. Any in Gn5 contains five linearly independent at least n+3 5-cyclic must equal to twice n+4. In this paper, minimum degree distance cyclic investigated. To find some transformations T have been d...

a set $wsubseteq v(g)$ is called a resolving set for $g$, if for each two distinct vertices $u,vin v(g)$ there exists $win w$ such that $d(u,w)neq d(v,w)$, where $d(x,y)$ is the distance between the vertices $x$ and $y$. the minimum cardinality of a resolving set for $g$ is called the metric dimension of $g$, and denoted by $dim(g)$. in this paper, it is proved that in a connected graph $...

The eccentricity of a vertex $v$ is the maximum distance between $v$ and anyother vertex. A vertex with maximum eccentricity is called a peripheral vertex.The peripheral Wiener index $ PW(G)$ of a graph $G$ is defined as the sum ofthe distances between all pairs of peripheral vertices of $G.$ In this paper, weinitiate the study of the peripheral Wiener index and we investigate its basicproperti...