Frank Harary and Allen J. Schwenk have given a formula for counting the number of non-isomorphic caterpillars on \(n\) vertices with \(n ≥ 3\). Inspired by the formula of Frank Harary and Allen
J. Schwenk, in this paper, we give a formula for counting the number of non-isomorphic caterpillars with the same degree sequence.

The line graph \(L(G)\) of a connected graph G, has vertex set identical with the set of edges of \(G\), and two vertices of \(L(G)\) are adjacent if and only if the corresponding edges are adjacent in \(G\). Ivan Gutman et al examined the dependency of certain physio-chemical properties of alkanes in boiling point, molar volume, and molar refraction, heat of vapourization, critical temperature, critical pressure and surface tension on the Bertz indices of \(L'(G)\) Dobrynin and Melnikov conjectured that there exists no nontrivial tree \(T\) and \(i≥3\), such that \(W(L'(T)) = W(T)\). In this paper we study Wiener and Zagreb indices for line graphs of Complete graph, Complete bipartite graph and wheel graph.

A set S of vertices in a graph G is called a dominating set of G if every vertex in V(G)\S is adjacent to some vertex in S. A set S is said to be a power dominating set of G if every vertex in the system is monitored by the set S following a set of rules for power system monitoring. The power domination number of G is the minimum cardinality of a power dominating set of G. In this paper, we solve the power domination number for certain nanotori such as H-Naphtelanic, \(C_5C_6C_7[m,n]\) nanotori and \(C_4C_6C_8[m,n]\) nanotori.

Let \(G_k, (k ≥ 0)\) be the family of graphs that have exactly k cycles. For \(0 ≤ k ≤ 3\), we compute the Hadwiger number for graphs in \(G_k\) and further deduce that the Hadwiger Conjecture is true for such families of graphs.

Split domination number of a graph is the cardinality of a minimum dominating set whose removal disconnects the graph. In this paper, we define a special family of Halin graphs and determine the split domination number of those graphs. We show that the construction yield non-isomorphic families of Halin graphs but with same split domination numbers.

A graph \(G(v,E)\) with \(n\) vertices is said to have modular multiplicative divisor bijection \(f: V(G)→{1,2,.., n}\) and the induced function \(f*: E(G) → {0,1,2,…, n – 1}\) where \(f*(uv)=f(u)f(v)(mod\,\,n)\) for all \(uv \in E(G)\) such that \(n\) divides the sum of all edge labels of \(G\). This paper studies MMD labeling of an even arbitrary supersubdivision (EASS) of corona related graphs.

In this paper, the distance and degree based topological indices for double silicate chain graph are obtained.

In this paper, we introduce a new form of fuzzy number named as Icosikaitetragonal fuzzy number with its membership function. It includes some basic arithmetic operations like addition, subtraction, multiplication and scalar multiplication by means of \(\alpha\)-cut with numerical illustrations.

In this paper, we determine the wirelength of embedding complete bipartite graphs \(K_{2^{n-1}, 2^{n-1}}\) into 1-rooted sibling tree \(ST_n^1\), and Cartesian product of 1-rooted sibling trees and paths.

A dominator coloring is a proper vertex coloring of a graph \(G\) such that each vertex is adjacent to all the vertices of at least one color class or either alone in its color class. The minimum cardinality among all dominator coloring of \(G\) is a dominator chromatic number of \(G\), denoted by \(X_d(G)\). On removal of a vertex the dominator chromatic number may increase or decrease or remain unaltered. In this paper, we have characterized nontrivial trees for which dominator chromatic number is stable.