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A double Roman dominating function (DRDF) on a graph G = (V , E) is a function f : V → { 0 , 1 , 2 , 3 } satisfying (i) if f (v) = 0 then there must be at least two neighbors assigned two under f or one neighbor w with f (w) = 3 ; and (ii) if f (v) = 1 then v must be adjacent to a vertex w such that f (w) ≥ 2. A DRDF is an outer-independent total double Roman dominating function (OITDRDF) on G if the set of vertices labeled 0 induces an edgeless subgraph and the subgraph induced by the vertices with a non-zero label has no isolated vertices. The weight of an OITDRDF is the sum of its function values over all vertices, and the outer-independent total Roman domination number γ tdR oi (G) is the minimum weight of an OITDRDF on G. In this paper, we establish various bounds on γ tdR oi (G). In particular, we present Nordhaus–Gaddum-type inequalities for this parameter. Some of our results improve the previous results. [ABSTRACT FROM AUTHOR]

For a graph G=(V,E), a triple Roman dominating function (3RD-function) is a function f:V→{0,1,2,3,4} having the property that (i) if f(v)=0 then v must have either one neighbor u with f(u)=4, or two neighbors u,w with f(u)+f(w)≥5 or three neighbors u,w,z with f(u)=f(w)=f(z)=2, (ii) if f(v)=1 then v must have one neighbor u with f(u)≥3 or two neighbors u,w with f(u)=f(w)=2, and (iii) if f(v)=2 then v must have one neighbor u with f(u)≥2. The weight of a 3RDF f is the sum f(V)=∑v∈Vf(v), and the minimum weight of a 3RD-function on G is the triple Roman domination number of G, denoted by γ[3R](G). In this paper, we prove that for any connected graph G of order n with minimum degree at least two, γ[3R](G)≤3n2. [ABSTRACT FROM AUTHOR]

A restrained Italian dominating function (RIDF) on a graph G = (V , E) is a function f : V → { 0 , 1 , 2 } satisfying: (i) f (N (u)) ≥ 2 for every vertex u ∈ V (G) with f (u) = 0 , where N (u) is the set of vertices adjacent to u ; (ii) the subgraph induced by the vertices assigned 0 under f has no isolated vertices. The weight of an RIDF is the sum of its function values over the whole set of vertices, and the restrained Italian domination number γ r I (G) is the minimum weight of an RIDF on G. In this paper, we initiate the study of the restrained Italian bondage number b r I (G) of a graph G with no isolated vertices defined as the smallest size of set of edges F ⊆ E (G) for which γ r I (G − F) > γ r I (G). We begin by showing that the decision problem associated with the restrained Italian bondage problem is NP-hard. Then basic properties of the restrained Italian bondage number are presented. Finally, some sharp bounds for b r I (G) are also established. [ABSTRACT FROM AUTHOR]

The modified Sombor index of a graph G is defined as the sum of weights 1 / d u 2 + d v 2 over all edges u v of G , where d u and d v are the degrees of the vertices u and v in G , respectively. In this paper, we give a sharp lower bound on the modified Sombor index of unicyclic graphs with given diameter in terms the number of vertices. [ABSTRACT FROM AUTHOR]

Let k be a positive integer. A restrained k -rainbow dominating function (RkRD-function) of a graph G = (V , E) is a function f from the vertex set V to the set of all subsets of the set { 1 , 2 , ... , k } satisfying: (i) for any vertex v ∈ V with f (v) = ∅ the condition ⋃ u ∈ N (v) f (u) = { 1 , 2 , ... , k } is fulfilled, where N (v) is the set of all vertices adjacent to v ; (ii) the subgraph induced by the vertices assigned ∅ under f has no isolated vertices. The weight of an RkRD-function f is the value ω (f) = ∑ v ∈ V | f (v) | , and the restrained k -rainbow domination number of G , denoted by γ r r k (G) , is the minimum weight of an RkRD-function of G. In this paper, we continue the study of the restrained k -rainbow reinforcement number r r r k (G) of a graph G defined as the cardinality of a smallest set of edges that we must add to G to decrease γ r r k (G). We shall first show that the decision problem associated with γ r r k (G) is NP-hard for arbitrary graphs. Then several properties as well as some sharp bounds of the restrained k -rainbow reinforcement number are investigated. [ABSTRACT FROM AUTHOR]

In chemistry, the energy of a graph is of interest since it can be used to approximate the total π-electron energy of molecules. The spectral moments of the edge adjacency matrix and adjacency matrix recently have been successfully employed in quantitative structure-property relationship (QSPR) and quantitative structure-activity relationship (QSAR) studies of alkanes, alkyl halides, benzyl alcohols, cycloalkanes, and benzenoid hydrocarbons. Let G be a molecular graph and L(G) be a line graph, then the eigenvalues of G and L(G) are denoted by, λ 1 ≥ λ 2 ≥ ... ≥ λ n and γ 1 ≥ γ 2 ≥ ... ≥ γ n , respectively. The energies A and L(G) are defined as E (G) = ∑ i = 1 n | λ i | and E (L (G)) = ∑ i = 1 n | γ i |. In this paper, we examined both energies with spectral moments of the edge adjacency matrix and adjacency matrix of benzenoid hydrocarbons. These theoretical conclusions provide practical guiding significance for pharmaceutical engineering, complex network and quantify the degree of folding of long organic molecules. [ABSTRACT FROM AUTHOR]

Let G be a graph of order n and d i be the degree of the vertex v i , for i = 1 , 2 , ... , n. The ISI matrix of G is the square matrix of order n whose (i , j) -entry is equal to d i d j d i + d j if v i is adjacent to v j , and zero otherwise. Let μ 1 ≥ μ 2 ≥ ⋯ ≥ μ n , be the eigenvalues of ISI matrix. The ISI energy of a graph G , denoted by ξ ISI (G) , is defined as the sum of the absolute values of the eigenvalues of ISI matrix. In this paper, we prove that the star has the minimum ISI energy among trees. [ABSTRACT FROM AUTHOR]

For a graph G = (V, E), a double Roman dominating function (DRDF) f: V → {0, 1, 2, 3} has the property that for every vertex v ∈ V with f(v)=0, either there exists a neighbor u ∈ N(v), with f(u)=3, or at least two neighbors x, y ∈ N(v) having f(x)=f(y)=2, and every vertex with value 1 under f has at least a neighbor with value 2 or 3. The weight of a DRDF is the sum f(V)=∑v ∈ Vf(v). A DRDF f is an independent double Roman dominating function (IDRDF) if the vertices with weight at least two form an independent set. The independent double Roman domination number idR(G) is the minimum weight of an IDRDF on G. In this paper, we show that for every tree T with diameter at least three, i(T)+iR(T)-(s(T))/2 + 1 ≤ idR(T)≤i(T)+iR(T)+s(T)-2, where i(T),iR(T) and s(T) are the independent domination number, the independent Roman domination number and the number of support vertex of T, respectively. [ABSTRACT FROM AUTHOR]

Let G be a molecular graph with n vertices, m edges and G(L) be a line graph. Both G and L can be represented by their adjacency matrices A and E, respectively. The eigenvalues of G and L are denoted by, λ 1 ⩾ λ 2 ⩾ ... ⩾ λ n and γ 1 ⩾ γ 2 ⩾ ... ⩾ γ n , respectively. The Estrada index A and L are defined as E E (G) = ∑ i = 1 n e λ i and E E (L) = ∑ i = 1 n e γ i . In this article, we present some new lower and upper bounds for the Estrada index molecular and line graphs of phenylenes. [ABSTRACT FROM AUTHOR]

The forgotten topological index (F -index) of a graph G , denoted by F (G) , is defined as the sum of cubes of the degrees of all the vertices in a graph. A connected graph with equal order and size is called unicyclic graph, where order is number of vertices and size is number of edges. In this paper, we investigate the new sharp upper bounds on F -index in terms of the order and maximum degree. [ABSTRACT FROM AUTHOR]

Let G be a graph with n vertices, and let d u be the degree of the vertex u in the graph G. The Randić matrix of G is the square matrix of order n whose (i , j) -entry is equal to (d u d v) − 1 2 if the vertex u and the vertex v of G are adjacent, and 0 otherwise. The Randić eigenvalues of G are the eigenvalues of its Randić matrix and the Randić energy of G is the sum of the absolute values of its Randić eigenvalues. In this paper, we obtain some new results for the Randić eigenvalues and the Randić energy of a graph. [ABSTRACT FROM AUTHOR]