Let [Formula: see text] be a simple, finite, undirected graph with [Formula: see text] vertices. ... more Let [Formula: see text] be a simple, finite, undirected graph with [Formula: see text] vertices. The main purpose of this paper introduces the concepts of the minimum covering Gutman Estrada index, the minimum covering Seidel Estrada index, the minimum covering distance Estrada index, the minimum covering Randić Estrada index, the minimum covering Harary Estrada index, and the minimum covering color Estrada index of a graph. First, we compute the new concepts for some of the graphs, such as cocktail party, star, crown, complete and complete bipartite. Moreover, we establish upper and lower bounds for the new concepts.
Let D = (V, A) be a finite simple directed graph (digraph). A function f : V −→ {−1, 1} is called... more Let D = (V, A) be a finite simple directed graph (digraph). A function f : V −→ {−1, 1} is called a twin signed k-dominating function (TSkDF) if f (N − [v]) ≥ k and f (N + [v]) ≥ k for each vertex v ∈ V. The twin signed k-domination number of D is γ * sk (D) = min{ω(f) | f is a TSkDF of D}. In this paper, we initiate the study of twin signed k-domination in digraphs and present some bounds on γ * sk (D) in terms of the order, size and maximum and minimum indegrees and outdegrees, generalising some of the existing bounds for the twin signed domination numbers in digraphs and the signed k-domination numbers in graphs. In addition, we determine the twin signed k-domination numbers of some classes of digraphs.
The sum-connectivity index of a graph G is defined as the sum of weights 1 / d u + d v over all e... more The sum-connectivity index of a graph G is defined as the sum of weights 1 / d u + d v over all edges u v of G , where d u and d v are the degrees of the vertices u and v in graph G , respectively. In this paper, we give a sharp lower bound on the sum-connectivity index unicyclic graphs of order n ≥ 7 and diameter D G ≥ 5 .
Let G be a graph with vertex set V(G) and edge set E(G). Let d u denote the degree of vertex u ∈ ... more Let G be a graph with vertex set V(G) and edge set E(G). Let d u denote the degree of vertex u ∈ V(G). e geometric-arithmetic index of G is defined as GA(G) � uv∈E(G) (2 ���� d u d v /(d u + d v)). In this paper, we obtain some new lower and upper bounds for the geometric-arithmetic index and improve some known bounds. Moreover, we investigate the relationships between geometricarithmetic index and several other topological indices.
In this paper, we provide some conditions for the existence of a coincidence point of single-valu... more In this paper, we provide some conditions for the existence of a coincidence point of single-valued and multivalued mappings involving generalized α − ψ-Geraghty contractions endowed with a graph. Our main results improve the existing results in the corresponding literature. We also present examples to support the obtained results.
Let G = (V, E) be a simple graph. A function f : V → {−1, 1} is called an inverse signed total do... more Let G = (V, E) be a simple graph. A function f : V → {−1, 1} is called an inverse signed total dominating function if the sum of its function values over any open neighborhood is at most zero. The inverse signed total domination number of G, denoted by γ 0 st (G), equals to the maximum weight of an inverse signed total dominating function of G. In this paper, we establish upper bounds on the inverse signed total domination number of graphs in terms of their order, size and maximum and minimum degrees.
Let G be a simple graph without isolated vertices with vertex set V (G) and edge set E(G) and let... more Let G be a simple graph without isolated vertices with vertex set V (G) and edge set E(G) and let k be a positive integer. A function f : E(G) −→ {−1, 1} is said to be a signed star k-dominating function on G if e∈E(v) f (e) ≥ k for every vertex v of G, where E(v) = {uv ∈ E(G) | u ∈ N (v)}. A set {f1, f2,. .. , f d } of signed star k-dominating functions on G with the property that d i=1 fi(e) ≤ k for each e ∈ E(G), is called a signed star (k, k)-dominating family (of functions) on G. The maximum number of functions in a signed star (k, k)-dominating family on G is the signed star (k, k)-domatic number of G, denoted by d (k,k) SS (G). In this paper we study properties of the signed star (k, k)-domatic number d (k,k) SS (G). In particular, we present bounds on d (k,k) SS (G), and we determine the signed (k, k)-domatic number of some regular graphs. Some of our results extend these given by Atapour, Sheikholeslami, Ghameslou and Volkmann [Signed star domatic number of a graph, Discrete Appl. Math. 158 (2010), 213-218] for the signed star domatic number.
In a graph G, a vertex dominates itself and its neighbors. A subset S ⊆ V (G) is a double dominat... more In a graph G, a vertex dominates itself and its neighbors. A subset S ⊆ V (G) is a double dominating set of G if S dominates every vertex of G at least twice. The double domination number dd(G) is the minimum cardinality of a double dominating set of G. The double domination subdivision number sd dd (G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the double domination number. In this paper first we establish upper bounds on the double domination subdivision number for arbitrary graphs in terms of vertex degree. Then we present several different conditions on G which are sufficient to imply that sd dd (G) 3. We also prove that 1 sd dd (T) 2 for every tree T, and characterize the trees T for which sd dd (T) = 2.
Let G be a simple graph without isolated vertices with vertex set V (G) and edge set E(G) and let... more Let G be a simple graph without isolated vertices with vertex set V (G) and edge set E(G) and let k be a positive integer. A function f : E(G) −→ {−1, 1} is said to be a signed star k-dominating function on G if e∈E(v) f (e) ≥ k for every vertex v of G, where E(v) = {uv ∈ E(G) | u ∈ N (v)}. A set {f1, f2,. .. , f d } of signed star k-dominating functions on G with the property that d i=1 fi(e) ≤ k for each e ∈ E(G), is called a signed star (k, k)-dominating family (of functions) on G. The maximum number of functions in a signed star (k, k)-dominating family on G is the signed star (k, k)-domatic number of G, denoted by d (k,k) SS (G). In this paper we study properties of the signed star (k, k)-domatic number d (k,k) SS (G). In particular, we present bounds on d (k,k) SS (G), and we determine the signed (k, k)-domatic number of some regular graphs. Some of our results extend these given by Atapour, Sheikholeslami, Ghameslou and Volkmann [Signed star domatic number of a graph, Discrete Appl. Math. 158 (2010), 213-218] for the signed star domatic number.
A b s t r ac t. Let D = (V, A) be a finite simple directed graph (shortly digraph). A function f ... more A b s t r ac t. Let D = (V, A) be a finite simple directed graph (shortly digraph). A function f : V −→ {−1, 1} is called a twin signed dominating function (TSDF) if f (N − [v]) 1 and f (N + [v]) 1 for each vertex v ∈ V. The twin signed domination number of D is γ * s (D) = min{ω(f) | f is a TSDF of D}. In this paper, we initiate the study of twin signed domination in digraphs and we present sharp lower bounds for γ * s (D) in terms of the order, size and maximum and minimum indegrees and outdegrees. Some of our results are extensions of well-known lower bounds of the classical signed domination numbers of graphs.
Let $G=(V,E)$ be a finite and simple graph of order $n$ and maximumdegree $\Delta$. A signed stro... more Let $G=(V,E)$ be a finite and simple graph of order $n$ and maximumdegree $\Delta$. A signed strong total Roman dominating function ona graph $G$ is a function $f:V(G)\rightarrow\{-1, 1,2,\ldots, \lceil\frac{\Delta}{2}\rceil+1\}$ satisfying the condition that (i) forevery vertex $v$ of $G$, $f(N(v))=\sum_{u\in N(v)}f(u)\geq 1$, where$N(v)$ is the open neighborhood of $v$ and (ii) every vertex $v$ forwhich $f(v)=-1$ is adjacent to at least one vertex$w$ for which $f(w)\geq 1+\lceil\frac{1}{2}\vert N(w)\cap V_{-1}\vert\rceil$, where$V_{-1}=\{v\in V: f(v)=-1\}$.The minimum of thevalues $\omega(f)=\sum_{v\in V}f(v)$, taken over all signed strongtotal Roman dominating functions $f$ of $G$, is called the signed strong totalRoman domination number of $G$ and is denoted by $\gamma_{ssTR}(G)$.In this paper, we initiate signed strong total Roman domination number of a graph and giveseveral bounds for this parameter. Then, among other results, we determine the signed strong total Roman dominat...
Let [Formula: see text] be a finite simple directed graph (shortly digraph). A function [Formula:... more Let [Formula: see text] be a finite simple directed graph (shortly digraph). A function [Formula: see text] is called a twin signed double Roman dominating function (TSDRDF) if (i) every vertex [Formula: see text] with [Formula: see text] has at least two in-neighbor assigned a 2 or at least an in-neighbor [Formula: see text] with [Formula: see text], also at least two out-neighbor assigned a 2 or at least an out-neighbor [Formula: see text] with [Formula: see text] (ii) every vertex [Formula: see text] with [Formula: see text] is adjacent to at least an in-neighbor and an out-neighbor [Formula: see text] with [Formula: see text] and (iii) [Formula: see text] and [Formula: see text] hold for any vertex [Formula: see text]. The weight of a TSDRDF [Formula: see text] is [Formula: see text], the twin signed double Roman domination number [Formula: see text] of [Formula: see text] is the minimum weight of a TSDRDF on [Formula: see text]. In this paper, we initiate the study of twin sign...
Let $D=(V,A)$ be a finite simple directed graph. A function$f:Vlongrightarrow {-1,0,1}$ is called... more Let $D=(V,A)$ be a finite simple directed graph. A function$f:Vlongrightarrow {-1,0,1}$ is called a twin minus dominatingfunction (TMDF) if $f(N^-[v])ge 1$ and $f(N^+[v])ge 1$ for eachvertex $vin V$. The twin minus domination number of $D$ is$gamma_{-}^*(D)=min{w(f)mid f mbox{ is a TMDF of } D}$. Inthis paper, we initiate the study of twin minus domination numbersin digraphs and present some lower bounds for $gamma_{-}^*(D)$ interms of the order, size and maximum and minimum in-degrees andout-degrees.
e Randić index of a graph G is defined as the sum of weights 1/ ���� d u d v over all edges uv of... more e Randić index of a graph G is defined as the sum of weights 1/ ���� d u d v over all edges uv of G, where d u and d v are the degrees of the vertices u and v in G, respectively. In this paper, we will obtain lower and upper bounds for the Randić index in terms of size, maximum degree, and minimum degree. Moreover, we obtain a generally lower and a general upper bound for the Randić index.
A function $f:V(G)rightarrow {-1,0,1}$ is a minus dominating function if for every vertex $vin... more A function $f:V(G)rightarrow {-1,0,1}$ is a minus dominating function if for every vertex $vin V(G)$, $sum_{uin N[v]}f(u)ge 1$. A minus dominating function $f$ of $G$ is called a global minus dominating function if $f$ is also a minus dominating function of the complement $overline{G}$ of $G$. The global minus domination number $gamma_{g}^-(G)$ of $G$ is defined as $gamma_{g}^-(G)=min{sum_{vin V(G)} f(v)mid f ;{rm is; a; global; minus; dominating; function}\ {rm of }; G}$. In this paper we initiate the study of the global minus domination number in graphs and we establish lower and upper bounds for the global minus domination number.
The ℱ -coindex (forgotten topological coindex) for a simple connected graph G is defined as the s... more The ℱ -coindex (forgotten topological coindex) for a simple connected graph G is defined as the sum of the terms ζ G 2 y + ζ G 2 x over all nonadjacent vertex pairs x , y of G , where ζ G y and ζ G x are the degrees of the vertices y and x in G , respectively. The ℱ -index of a graph is defined as the sum of cubes of the vertex degrees of the graph. This was introduced in 1972 in the same paper where the first and second Zagreb indices were introduced to study the structure dependency of total π -electron energy. Therefore, considering the importance of the ℱ -index and ℱ -coindex, in this paper, we study these indices, and we present new bounds for the ℱ -index and ℱ -coindex.
This research intends to investigate the existence results for both coincidence points and common... more This research intends to investigate the existence results for both coincidence points and common fixed point of generalized Geraghty multi-valued mappings endowed with a directed graph. The proven results are supported by an example. We also consider fractional integral equations as an application.
Let D = (V;A) be a finite simple directed graph (digraph). A function f : V ? {-1,1} is called a ... more Let D = (V;A) be a finite simple directed graph (digraph). A function f : V ? {-1,1} is called a twin signed k-dominating function (TSkDF) if f (N-[v]) ? k and f (N+[v]) ? k for each vertex v ? V. The twin signed k-domination number of D is ?* sk(D) = min{?(f)?f is a TSkDF of D}. In this paper, we initiate the study of twin signed k-domination in digraphs and present some bounds on ?* sk(D) in terms of the order, size and maximum and minimum indegrees and outdegrees, generalising some of the existing bounds for the twin signed domination numbers in digraphs and the signed k-domination numbers in graphs. In addition, we determine the twin signed k-domination numbers of some classes of digraphs.
Let D = (V, A) be a finite simple directed graph (shortly, digraph). A function f : V −→ {−1, 0, ... more Let D = (V, A) be a finite simple directed graph (shortly, digraph). A function f : V −→ {−1, 0, 1} is called a twin minus total dominating function (TMTDF) if f (N − (v)) ≥ 1 and f (N + (v)) ≥ 1 for each vertex v ∈ V. The twin minus total domination number of D is γ * mt (D) = min{w(f) | f is a TMTDF of D}. In this paper, we initiate the study of twin minus total domination numbers in digraphs and we present some lower bounds for γ * mt (D) in terms of the order, size and maximum and minimum in-degrees and out-degrees. In addition, we determine the twin minus total domination numbers of some classes of digraphs.
Let [Formula: see text] be a simple, finite, undirected graph with [Formula: see text] vertices. ... more Let [Formula: see text] be a simple, finite, undirected graph with [Formula: see text] vertices. The main purpose of this paper introduces the concepts of the minimum covering Gutman Estrada index, the minimum covering Seidel Estrada index, the minimum covering distance Estrada index, the minimum covering Randić Estrada index, the minimum covering Harary Estrada index, and the minimum covering color Estrada index of a graph. First, we compute the new concepts for some of the graphs, such as cocktail party, star, crown, complete and complete bipartite. Moreover, we establish upper and lower bounds for the new concepts.
Let D = (V, A) be a finite simple directed graph (digraph). A function f : V −→ {−1, 1} is called... more Let D = (V, A) be a finite simple directed graph (digraph). A function f : V −→ {−1, 1} is called a twin signed k-dominating function (TSkDF) if f (N − [v]) ≥ k and f (N + [v]) ≥ k for each vertex v ∈ V. The twin signed k-domination number of D is γ * sk (D) = min{ω(f) | f is a TSkDF of D}. In this paper, we initiate the study of twin signed k-domination in digraphs and present some bounds on γ * sk (D) in terms of the order, size and maximum and minimum indegrees and outdegrees, generalising some of the existing bounds for the twin signed domination numbers in digraphs and the signed k-domination numbers in graphs. In addition, we determine the twin signed k-domination numbers of some classes of digraphs.
The sum-connectivity index of a graph G is defined as the sum of weights 1 / d u + d v over all e... more The sum-connectivity index of a graph G is defined as the sum of weights 1 / d u + d v over all edges u v of G , where d u and d v are the degrees of the vertices u and v in graph G , respectively. In this paper, we give a sharp lower bound on the sum-connectivity index unicyclic graphs of order n ≥ 7 and diameter D G ≥ 5 .
Let G be a graph with vertex set V(G) and edge set E(G). Let d u denote the degree of vertex u ∈ ... more Let G be a graph with vertex set V(G) and edge set E(G). Let d u denote the degree of vertex u ∈ V(G). e geometric-arithmetic index of G is defined as GA(G) � uv∈E(G) (2 ���� d u d v /(d u + d v)). In this paper, we obtain some new lower and upper bounds for the geometric-arithmetic index and improve some known bounds. Moreover, we investigate the relationships between geometricarithmetic index and several other topological indices.
In this paper, we provide some conditions for the existence of a coincidence point of single-valu... more In this paper, we provide some conditions for the existence of a coincidence point of single-valued and multivalued mappings involving generalized α − ψ-Geraghty contractions endowed with a graph. Our main results improve the existing results in the corresponding literature. We also present examples to support the obtained results.
Let G = (V, E) be a simple graph. A function f : V → {−1, 1} is called an inverse signed total do... more Let G = (V, E) be a simple graph. A function f : V → {−1, 1} is called an inverse signed total dominating function if the sum of its function values over any open neighborhood is at most zero. The inverse signed total domination number of G, denoted by γ 0 st (G), equals to the maximum weight of an inverse signed total dominating function of G. In this paper, we establish upper bounds on the inverse signed total domination number of graphs in terms of their order, size and maximum and minimum degrees.
Let G be a simple graph without isolated vertices with vertex set V (G) and edge set E(G) and let... more Let G be a simple graph without isolated vertices with vertex set V (G) and edge set E(G) and let k be a positive integer. A function f : E(G) −→ {−1, 1} is said to be a signed star k-dominating function on G if e∈E(v) f (e) ≥ k for every vertex v of G, where E(v) = {uv ∈ E(G) | u ∈ N (v)}. A set {f1, f2,. .. , f d } of signed star k-dominating functions on G with the property that d i=1 fi(e) ≤ k for each e ∈ E(G), is called a signed star (k, k)-dominating family (of functions) on G. The maximum number of functions in a signed star (k, k)-dominating family on G is the signed star (k, k)-domatic number of G, denoted by d (k,k) SS (G). In this paper we study properties of the signed star (k, k)-domatic number d (k,k) SS (G). In particular, we present bounds on d (k,k) SS (G), and we determine the signed (k, k)-domatic number of some regular graphs. Some of our results extend these given by Atapour, Sheikholeslami, Ghameslou and Volkmann [Signed star domatic number of a graph, Discrete Appl. Math. 158 (2010), 213-218] for the signed star domatic number.
In a graph G, a vertex dominates itself and its neighbors. A subset S ⊆ V (G) is a double dominat... more In a graph G, a vertex dominates itself and its neighbors. A subset S ⊆ V (G) is a double dominating set of G if S dominates every vertex of G at least twice. The double domination number dd(G) is the minimum cardinality of a double dominating set of G. The double domination subdivision number sd dd (G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the double domination number. In this paper first we establish upper bounds on the double domination subdivision number for arbitrary graphs in terms of vertex degree. Then we present several different conditions on G which are sufficient to imply that sd dd (G) 3. We also prove that 1 sd dd (T) 2 for every tree T, and characterize the trees T for which sd dd (T) = 2.
Let G be a simple graph without isolated vertices with vertex set V (G) and edge set E(G) and let... more Let G be a simple graph without isolated vertices with vertex set V (G) and edge set E(G) and let k be a positive integer. A function f : E(G) −→ {−1, 1} is said to be a signed star k-dominating function on G if e∈E(v) f (e) ≥ k for every vertex v of G, where E(v) = {uv ∈ E(G) | u ∈ N (v)}. A set {f1, f2,. .. , f d } of signed star k-dominating functions on G with the property that d i=1 fi(e) ≤ k for each e ∈ E(G), is called a signed star (k, k)-dominating family (of functions) on G. The maximum number of functions in a signed star (k, k)-dominating family on G is the signed star (k, k)-domatic number of G, denoted by d (k,k) SS (G). In this paper we study properties of the signed star (k, k)-domatic number d (k,k) SS (G). In particular, we present bounds on d (k,k) SS (G), and we determine the signed (k, k)-domatic number of some regular graphs. Some of our results extend these given by Atapour, Sheikholeslami, Ghameslou and Volkmann [Signed star domatic number of a graph, Discrete Appl. Math. 158 (2010), 213-218] for the signed star domatic number.
A b s t r ac t. Let D = (V, A) be a finite simple directed graph (shortly digraph). A function f ... more A b s t r ac t. Let D = (V, A) be a finite simple directed graph (shortly digraph). A function f : V −→ {−1, 1} is called a twin signed dominating function (TSDF) if f (N − [v]) 1 and f (N + [v]) 1 for each vertex v ∈ V. The twin signed domination number of D is γ * s (D) = min{ω(f) | f is a TSDF of D}. In this paper, we initiate the study of twin signed domination in digraphs and we present sharp lower bounds for γ * s (D) in terms of the order, size and maximum and minimum indegrees and outdegrees. Some of our results are extensions of well-known lower bounds of the classical signed domination numbers of graphs.
Let $G=(V,E)$ be a finite and simple graph of order $n$ and maximumdegree $\Delta$. A signed stro... more Let $G=(V,E)$ be a finite and simple graph of order $n$ and maximumdegree $\Delta$. A signed strong total Roman dominating function ona graph $G$ is a function $f:V(G)\rightarrow\{-1, 1,2,\ldots, \lceil\frac{\Delta}{2}\rceil+1\}$ satisfying the condition that (i) forevery vertex $v$ of $G$, $f(N(v))=\sum_{u\in N(v)}f(u)\geq 1$, where$N(v)$ is the open neighborhood of $v$ and (ii) every vertex $v$ forwhich $f(v)=-1$ is adjacent to at least one vertex$w$ for which $f(w)\geq 1+\lceil\frac{1}{2}\vert N(w)\cap V_{-1}\vert\rceil$, where$V_{-1}=\{v\in V: f(v)=-1\}$.The minimum of thevalues $\omega(f)=\sum_{v\in V}f(v)$, taken over all signed strongtotal Roman dominating functions $f$ of $G$, is called the signed strong totalRoman domination number of $G$ and is denoted by $\gamma_{ssTR}(G)$.In this paper, we initiate signed strong total Roman domination number of a graph and giveseveral bounds for this parameter. Then, among other results, we determine the signed strong total Roman dominat...
Let [Formula: see text] be a finite simple directed graph (shortly digraph). A function [Formula:... more Let [Formula: see text] be a finite simple directed graph (shortly digraph). A function [Formula: see text] is called a twin signed double Roman dominating function (TSDRDF) if (i) every vertex [Formula: see text] with [Formula: see text] has at least two in-neighbor assigned a 2 or at least an in-neighbor [Formula: see text] with [Formula: see text], also at least two out-neighbor assigned a 2 or at least an out-neighbor [Formula: see text] with [Formula: see text] (ii) every vertex [Formula: see text] with [Formula: see text] is adjacent to at least an in-neighbor and an out-neighbor [Formula: see text] with [Formula: see text] and (iii) [Formula: see text] and [Formula: see text] hold for any vertex [Formula: see text]. The weight of a TSDRDF [Formula: see text] is [Formula: see text], the twin signed double Roman domination number [Formula: see text] of [Formula: see text] is the minimum weight of a TSDRDF on [Formula: see text]. In this paper, we initiate the study of twin sign...
Let $D=(V,A)$ be a finite simple directed graph. A function$f:Vlongrightarrow {-1,0,1}$ is called... more Let $D=(V,A)$ be a finite simple directed graph. A function$f:Vlongrightarrow {-1,0,1}$ is called a twin minus dominatingfunction (TMDF) if $f(N^-[v])ge 1$ and $f(N^+[v])ge 1$ for eachvertex $vin V$. The twin minus domination number of $D$ is$gamma_{-}^*(D)=min{w(f)mid f mbox{ is a TMDF of } D}$. Inthis paper, we initiate the study of twin minus domination numbersin digraphs and present some lower bounds for $gamma_{-}^*(D)$ interms of the order, size and maximum and minimum in-degrees andout-degrees.
e Randić index of a graph G is defined as the sum of weights 1/ ���� d u d v over all edges uv of... more e Randić index of a graph G is defined as the sum of weights 1/ ���� d u d v over all edges uv of G, where d u and d v are the degrees of the vertices u and v in G, respectively. In this paper, we will obtain lower and upper bounds for the Randić index in terms of size, maximum degree, and minimum degree. Moreover, we obtain a generally lower and a general upper bound for the Randić index.
A function $f:V(G)rightarrow {-1,0,1}$ is a minus dominating function if for every vertex $vin... more A function $f:V(G)rightarrow {-1,0,1}$ is a minus dominating function if for every vertex $vin V(G)$, $sum_{uin N[v]}f(u)ge 1$. A minus dominating function $f$ of $G$ is called a global minus dominating function if $f$ is also a minus dominating function of the complement $overline{G}$ of $G$. The global minus domination number $gamma_{g}^-(G)$ of $G$ is defined as $gamma_{g}^-(G)=min{sum_{vin V(G)} f(v)mid f ;{rm is; a; global; minus; dominating; function}\ {rm of }; G}$. In this paper we initiate the study of the global minus domination number in graphs and we establish lower and upper bounds for the global minus domination number.
The ℱ -coindex (forgotten topological coindex) for a simple connected graph G is defined as the s... more The ℱ -coindex (forgotten topological coindex) for a simple connected graph G is defined as the sum of the terms ζ G 2 y + ζ G 2 x over all nonadjacent vertex pairs x , y of G , where ζ G y and ζ G x are the degrees of the vertices y and x in G , respectively. The ℱ -index of a graph is defined as the sum of cubes of the vertex degrees of the graph. This was introduced in 1972 in the same paper where the first and second Zagreb indices were introduced to study the structure dependency of total π -electron energy. Therefore, considering the importance of the ℱ -index and ℱ -coindex, in this paper, we study these indices, and we present new bounds for the ℱ -index and ℱ -coindex.
This research intends to investigate the existence results for both coincidence points and common... more This research intends to investigate the existence results for both coincidence points and common fixed point of generalized Geraghty multi-valued mappings endowed with a directed graph. The proven results are supported by an example. We also consider fractional integral equations as an application.
Let D = (V;A) be a finite simple directed graph (digraph). A function f : V ? {-1,1} is called a ... more Let D = (V;A) be a finite simple directed graph (digraph). A function f : V ? {-1,1} is called a twin signed k-dominating function (TSkDF) if f (N-[v]) ? k and f (N+[v]) ? k for each vertex v ? V. The twin signed k-domination number of D is ?* sk(D) = min{?(f)?f is a TSkDF of D}. In this paper, we initiate the study of twin signed k-domination in digraphs and present some bounds on ?* sk(D) in terms of the order, size and maximum and minimum indegrees and outdegrees, generalising some of the existing bounds for the twin signed domination numbers in digraphs and the signed k-domination numbers in graphs. In addition, we determine the twin signed k-domination numbers of some classes of digraphs.
Let D = (V, A) be a finite simple directed graph (shortly, digraph). A function f : V −→ {−1, 0, ... more Let D = (V, A) be a finite simple directed graph (shortly, digraph). A function f : V −→ {−1, 0, 1} is called a twin minus total dominating function (TMTDF) if f (N − (v)) ≥ 1 and f (N + (v)) ≥ 1 for each vertex v ∈ V. The twin minus total domination number of D is γ * mt (D) = min{w(f) | f is a TMTDF of D}. In this paper, we initiate the study of twin minus total domination numbers in digraphs and we present some lower bounds for γ * mt (D) in terms of the order, size and maximum and minimum in-degrees and out-degrees. In addition, we determine the twin minus total domination numbers of some classes of digraphs.
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