A bstract. We study cellular automata. with addi tive rules on finite undi rected graphs. The addition is carried out in some finite abelian mono id. We investigate the prob lem of deciding wheth er a given configuration has a predecessor . Depending on th e underlying monoid this prob lem is solvable in polynomial time or NP-complete. Furt hermo re, we st udy the global reversibility of cellul...

The sum-connectivity index R′(G) of a graph G is the sum of the weights (du +dv) − 2 of all edges uv of G, where du and dv are the degrees of the vertices u and v in G. This index was recently introduced in [B. Zhou, N. Trinajstić, On a novel connectivity index, J. Math. Chem. 46(2009), 1252–1270]. In this paper, we give the sharp lower bound of the sum-connectivity index of n-vertex unicyclic ...

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...

Let $G$ be a simple graph on $n$ vertices. $L_G \text{ and } \mathcal{I}_G \: $ denote the Lov\'asz-Saks-Schrijver(LSS) ideal parity binomial edge of in polynomial ring $S = \mathbb{K}[x_1,\ldots, x_n, y_1, \ldots, y_n] respectively. We classify graphs whose LSS ideals are complete intersections. also almost intersections, we prove that their Rees algebra is Cohen-Macaulay. compute second grade...

In a connected graph G , the cardinality of smallest ordered set vertices that distinguishes every element V ( ) ? E is called mixed metric dimension . this paper we first establish exact value unicyclic which derived from structure We further consider graphs with edge disjoint cycles, where for each cycle C i define subgraph in only cycle. Applying result to then yields such The obtained formu...

Given a graph G = ( V , E ) and an integer k ? 1 - hop dominating set D of is subset such that, for every vertex v ? there exists node u whose distance from at most . A -hop minimum cardinality called In this paper, we present linear-time algorithm that finds in cactus graphs which improves the O n 3 -time Borradaile Le (2017). To achieve this, show problem unicyclic reduces to piercing circula...

Abstract Let G be a graph. Assume that to each vertex of set vertices $S\subseteq V(G)$ robot is assigned. At stage one can move neighbouring vertex. Then S mobile general position if there exists sequence moves the robots such all are visited while maintaining property at times. The number cardinality largest . We give bounds on and determine exact values for certain common classes graphs, inc...

-A vertex v V(G) is said to be a self vertex switching of G if G is isomorphic to G v , where G v is the graph obtained from G by deleting all edges of G incident to v and adding all edges incident to v which are not in G. In [5], trees and forests are characterized, each with a self vertex switching. In [6], connected unicyclic graphs, each with a self vertex switching are characterized. In th...

a 2-emph{rainbow dominating function} (2rdf) on a graph $g=(v, e)$ is afunction $f$ from the vertex set $v$ to the set of all subsets of the set${1,2}$ such that for any vertex $vin v$ with $f(v)=emptyset$ thecondition $bigcup_{uin n(v)}f(u)={1,2}$ is fulfilled. a 2rdf $f$ isindependent (i2rdf) if no two vertices assigned nonempty sets are adjacent.the emph{weight} of a 2rdf $f$ is the value $o...

Given a graph G, the burning number of G is smallest integer k for which there are vertices $$x_1, x_2,\ldots ,x_k$$ such that $$(x_1,x_2,\ldots ,x_k)$$ sequence G. It has been shown problem NP-complete, even trees with maximum degree three, or linear forests. A t-unicyclic unicycle in unique vertex greater than two $$ t + 2 . In this paper, we first present bounds graphs, and then use numbers ...