Let G = (V, E) be a graph with chromatic number χ(G). A dominating set D of G is called a chromatic transversal dominating set (ctd-set) if D intersects every color class of every χ-partition of G. The minimum cardinality of a ctd-set of G is called the chromatic transversal domination number of G and is denoted by γct(G). In this paper we characterize the class of trees, unicyclic graphs and c...

The algebraic connectivity ofG is the second smallest eigenvalue of its Laplacianmatrix. Let Un be the set of all unicyclic graphs of order n. In this paper, wewill provide the ordering of unicyclic graphs inUn up to the last seven graphs according to their algebraic connectivities when n ≥ 13. This extends the results of Liu and Liu [Y. Liu, Y. Liu, The ordering of unicyclic graphswith the sma...

Castelnuovo–Mumford regularity is one of the most important invariants in commutative algebra. It has been investigated by various authors from different point of view. One of the interesting properties of regularity is its asymptotic behavior. In this talk first we will see some well-known facts of regularity and then I will present some known results of asymptotic behavior of quadratic square...

A graph G of order n is called arbitrarily vertex decomposable if for each sequence (n1, . . . , nk) of positive integers such that ∑k i=1 ni = n, there exists a partition (V1, . . . , Vk) of vertex set of G such that for every i ∈ {1, . . . , k} the set Vi induces a connected subgraph of G on ni vertices. We consider arbitrarily vertex decomposable unicyclic graphs with dominating cycle. We al...

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...