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MATHEMATICS (818 journals)                  1 2 3 4 5 | Last

Showing 1 - 200 of 538 Journals sorted alphabetically
Abakós     Open Access   (Followers: 5)
Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg     Hybrid Journal   (Followers: 3)
Academic Voices : A Multidisciplinary Journal     Open Access   (Followers: 2)
Accounting Perspectives     Full-text available via subscription   (Followers: 8)
ACM Transactions on Algorithms (TALG)     Hybrid Journal   (Followers: 17)
ACM Transactions on Computational Logic (TOCL)     Hybrid Journal   (Followers: 5)
ACM Transactions on Mathematical Software (TOMS)     Hybrid Journal   (Followers: 9)
ACS Applied Materials & Interfaces     Hybrid Journal   (Followers: 46)
Acta Applicandae Mathematicae     Hybrid Journal   (Followers: 1)
Acta Mathematica     Hybrid Journal   (Followers: 11)
Acta Mathematica Hungarica     Hybrid Journal   (Followers: 2)
Acta Mathematica Scientia     Full-text available via subscription   (Followers: 5)
Acta Mathematica Sinica, English Series     Hybrid Journal   (Followers: 6)
Acta Mathematica Vietnamica     Hybrid Journal  
Acta Mathematicae Applicatae Sinica, English Series     Hybrid Journal  
Advanced Science Letters     Full-text available via subscription   (Followers: 13)
Advances in Applied Clifford Algebras     Hybrid Journal   (Followers: 5)
Advances in Calculus of Variations     Hybrid Journal   (Followers: 7)
Advances in Catalysis     Full-text available via subscription   (Followers: 8)
Advances in Complex Systems     Hybrid Journal   (Followers: 10)
Advances in Computational Mathematics     Hybrid Journal   (Followers: 22)
Advances in Decision Sciences     Open Access   (Followers: 4)
Advances in Difference Equations     Open Access   (Followers: 4)
Advances in Fixed Point Theory     Open Access   (Followers: 9)
Advances in Geosciences (ADGEO)     Open Access   (Followers: 21)
Advances in Linear Algebra & Matrix Theory     Open Access   (Followers: 10)
Advances in Materials Science     Open Access   (Followers: 21)
Advances in Mathematical Physics     Open Access   (Followers: 8)
Advances in Mathematics     Full-text available via subscription   (Followers: 19)
Advances in Numerical Analysis     Open Access   (Followers: 8)
Advances in Operations Research     Open Access   (Followers: 14)
Advances in Operator Theory     Hybrid Journal   (Followers: 4)
Advances in Porous Media     Full-text available via subscription   (Followers: 6)
Advances in Pure and Applied Mathematics     Hybrid Journal   (Followers: 10)
Advances in Pure Mathematics     Open Access   (Followers: 10)
Advances in Science and Research (ASR)     Open Access   (Followers: 8)
Aequationes Mathematicae     Hybrid Journal   (Followers: 2)
African Journal of Educational Studies in Mathematics and Sciences     Full-text available via subscription   (Followers: 12)
African Journal of Mathematics and Computer Science Research     Open Access   (Followers: 7)
Afrika Matematika     Hybrid Journal   (Followers: 3)
Air, Soil & Water Research     Open Access   (Followers: 13)
AKSIOMA Journal of Mathematics Education     Open Access   (Followers: 3)
AKSIOMATIK : Jurnal Penelitian Pendidikan dan Pembelajaran Matematika     Open Access   (Followers: 1)
Al-Jabar : Jurnal Pendidikan Matematika     Open Access   (Followers: 1)
Al-Qadisiyah Journal for Computer Science and Mathematics     Open Access   (Followers: 1)
AL-Rafidain Journal of Computer Sciences and Mathematics     Open Access   (Followers: 6)
Algebra and Logic     Hybrid Journal   (Followers: 8)
Algebra Colloquium     Hybrid Journal   (Followers: 4)
Algebra Universalis     Hybrid Journal   (Followers: 2)
Algorithmic Operations Research     Open Access   (Followers: 5)
Algorithms     Open Access   (Followers: 13)
Algorithms Research     Open Access   (Followers: 1)
American Journal of Computational and Applied Mathematics     Open Access   (Followers: 10)
American Journal of Mathematical Analysis     Open Access   (Followers: 2)
American Journal of Mathematical and Management Sciences     Hybrid Journal   (Followers: 1)
American Journal of Mathematics     Full-text available via subscription   (Followers: 7)
American Journal of Operations Research     Open Access   (Followers: 8)
American Mathematical Monthly     Full-text available via subscription   (Followers: 7)
An International Journal of Optimization and Control: Theories & Applications     Open Access   (Followers: 12)
Analele Universitatii Ovidius Constanta - Seria Matematica     Open Access  
Analysis and Applications     Hybrid Journal   (Followers: 1)
Analysis and Mathematical Physics     Hybrid Journal   (Followers: 7)
Analysis Mathematica     Full-text available via subscription  
Analysis. International mathematical journal of analysis and its applications     Hybrid Journal   (Followers: 5)
Anargya : Jurnal Ilmiah Pendidikan Matematika     Open Access   (Followers: 8)
Annales Mathematicae Silesianae     Open Access   (Followers: 2)
Annales mathématiques du Québec     Hybrid Journal   (Followers: 4)
Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica     Open Access   (Followers: 1)
Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica     Open Access  
Annali di Matematica Pura ed Applicata     Hybrid Journal   (Followers: 1)
Annals of Combinatorics     Hybrid Journal   (Followers: 3)
Annals of Data Science     Hybrid Journal   (Followers: 16)
Annals of Discrete Mathematics     Full-text available via subscription   (Followers: 8)
Annals of Functional Analysis     Hybrid Journal   (Followers: 3)
Annals of Mathematics     Full-text available via subscription   (Followers: 1)
Annals of Mathematics and Artificial Intelligence     Hybrid Journal   (Followers: 15)
Annals of PDE     Hybrid Journal  
Annals of Pure and Applied Logic     Open Access   (Followers: 6)
Annals of the Alexandru Ioan Cuza University - Mathematics     Open Access  
Annals of the Institute of Statistical Mathematics     Hybrid Journal   (Followers: 1)
Annals of West University of Timisoara - Mathematics     Open Access  
Annals of West University of Timisoara - Mathematics and Computer Science     Open Access   (Followers: 2)
Annuaire du Collège de France     Open Access   (Followers: 6)
ANZIAM Journal     Open Access   (Followers: 1)
Applicable Algebra in Engineering, Communication and Computing     Hybrid Journal   (Followers: 3)
Applications of Mathematics     Hybrid Journal   (Followers: 2)
Applied Categorical Structures     Hybrid Journal   (Followers: 4)
Applied Computational Intelligence and Soft Computing     Open Access   (Followers: 15)
Applied Mathematics     Open Access   (Followers: 8)
Applied Mathematics     Open Access   (Followers: 4)
Applied Mathematics & Optimization     Hybrid Journal   (Followers: 12)
Applied Mathematics - A Journal of Chinese Universities     Hybrid Journal   (Followers: 2)
Applied Mathematics and Nonlinear Sciences     Open Access   (Followers: 1)
Applied Mathematics Letters     Full-text available via subscription   (Followers: 3)
Applied Mathematics Research eXpress     Hybrid Journal   (Followers: 2)
Applied Network Science     Open Access   (Followers: 3)
Applied Numerical Mathematics     Hybrid Journal   (Followers: 6)
Applied Spatial Analysis and Policy     Hybrid Journal   (Followers: 6)
Arab Journal of Mathematical Sciences     Open Access   (Followers: 4)
Arabian Journal of Mathematics     Open Access   (Followers: 2)
Archive for Mathematical Logic     Hybrid Journal   (Followers: 4)
Archive of Applied Mechanics     Hybrid Journal   (Followers: 6)
Archive of Numerical Software     Open Access  
Archives of Computational Methods in Engineering     Hybrid Journal   (Followers: 6)
Arkiv för Matematik     Hybrid Journal   (Followers: 1)
Armenian Journal of Mathematics     Open Access   (Followers: 1)
Arnold Mathematical Journal     Hybrid Journal   (Followers: 1)
Artificial Satellites     Open Access   (Followers: 23)
Asia-Pacific Journal of Operational Research     Hybrid Journal   (Followers: 3)
Asian Journal of Algebra     Open Access   (Followers: 1)
Asian Research Journal of Mathematics     Open Access  
Asian-European Journal of Mathematics     Hybrid Journal   (Followers: 3)
Australian Mathematics Teacher, The     Full-text available via subscription   (Followers: 7)
Australian Primary Mathematics Classroom     Full-text available via subscription   (Followers: 7)
Australian Senior Mathematics Journal     Full-text available via subscription   (Followers: 2)
Automatic Documentation and Mathematical Linguistics     Hybrid Journal   (Followers: 5)
Axioms     Open Access   (Followers: 1)
Baltic International Yearbook of Cognition, Logic and Communication     Open Access   (Followers: 2)
Banach Journal of Mathematical Analysis     Hybrid Journal   (Followers: 1)
Basin Research     Hybrid Journal   (Followers: 5)
BIBECHANA     Open Access   (Followers: 2)
Biomath     Open Access  
BIT Numerical Mathematics     Hybrid Journal   (Followers: 1)
Boletim Cearense de Educação e História da Matemática     Open Access  
Boletim de Educação Matemática     Open Access  
Boletín de la Sociedad Matemática Mexicana     Hybrid Journal  
Bollettino dell'Unione Matematica Italiana     Full-text available via subscription   (Followers: 3)
British Journal for the History of Mathematics     Hybrid Journal  
British Journal of Mathematical and Statistical Psychology     Full-text available via subscription   (Followers: 19)
Bruno Pini Mathematical Analysis Seminar     Open Access  
Buletinul Academiei de Stiinte a Republicii Moldova. Matematica     Open Access   (Followers: 14)
Bulletin des Sciences Mathamatiques     Full-text available via subscription   (Followers: 4)
Bulletin of Dnipropetrovsk University. Series : Communications in Mathematical Modeling and Differential Equations Theory     Open Access   (Followers: 3)
Bulletin of Mathematical Sciences     Open Access   (Followers: 1)
Bulletin of Symbolic Logic     Full-text available via subscription   (Followers: 3)
Bulletin of Taras Shevchenko National University of Kyiv. Series: Physics and Mathematics     Open Access   (Followers: 1)
Bulletin of the Australian Mathematical Society     Full-text available via subscription   (Followers: 2)
Bulletin of the Brazilian Mathematical Society, New Series     Hybrid Journal  
Bulletin of the Iranian Mathematical Society     Hybrid Journal  
Bulletin of the London Mathematical Society     Hybrid Journal   (Followers: 3)
Bulletin of the Malaysian Mathematical Sciences Society     Hybrid Journal  
Cadernos do IME : Série Matemática     Open Access   (Followers: 1)
Calculus of Variations and Partial Differential Equations     Hybrid Journal  
Canadian Journal of Mathematics / Journal canadien de mathématiques     Hybrid Journal  
Canadian Journal of Science, Mathematics and Technology Education     Hybrid Journal   (Followers: 22)
Canadian Mathematical Bulletin     Hybrid Journal  
Carpathian Mathematical Publications     Open Access   (Followers: 1)
Catalysis in Industry     Hybrid Journal   (Followers: 1)
CEAS Space Journal     Hybrid Journal   (Followers: 4)
CHANCE     Hybrid Journal   (Followers: 5)
Chaos, Solitons & Fractals     Hybrid Journal   (Followers: 3)
Chaos, Solitons & Fractals : X     Open Access   (Followers: 1)
ChemSusChem     Hybrid Journal   (Followers: 8)
Chinese Annals of Mathematics, Series B     Hybrid Journal  
Chinese Journal of Catalysis     Full-text available via subscription   (Followers: 3)
Chinese Journal of Mathematics     Open Access  
Ciencia     Open Access   (Followers: 1)
Clean Air Journal     Full-text available via subscription   (Followers: 1)
CODEE Journal     Open Access   (Followers: 3)
Cogent Mathematics     Open Access   (Followers: 2)
Cognitive Computation     Hybrid Journal   (Followers: 3)
Collectanea Mathematica     Hybrid Journal  
College Mathematics Journal     Hybrid Journal   (Followers: 4)
COMBINATORICA     Hybrid Journal  
Combinatorics, Probability and Computing     Hybrid Journal   (Followers: 4)
Combustion Theory and Modelling     Hybrid Journal   (Followers: 16)
Commentarii Mathematici Helvetici     Hybrid Journal  
Communications in Advanced Mathematical Sciences     Open Access  
Communications in Combinatorics and Optimization     Open Access  
Communications in Contemporary Mathematics     Hybrid Journal  
Communications in Mathematical Physics     Hybrid Journal   (Followers: 4)
Communications On Pure & Applied Mathematics     Hybrid Journal   (Followers: 4)
Complex Analysis and its Synergies     Open Access   (Followers: 3)
Complex Variables and Elliptic Equations: An International Journal     Hybrid Journal  
Composite Materials Series     Full-text available via subscription   (Followers: 11)
Compositio Mathematica     Full-text available via subscription  
Comptes Rendus Mathematique     Full-text available via subscription  
Computational and Applied Mathematics     Hybrid Journal   (Followers: 4)
Computational and Mathematical Methods     Hybrid Journal  
Computational and Mathematical Methods in Medicine     Open Access   (Followers: 3)
Computational and Mathematical Organization Theory     Hybrid Journal   (Followers: 1)
Computational Complexity     Hybrid Journal   (Followers: 4)
Computational Mathematics and Modeling     Hybrid Journal   (Followers: 9)
Computational Mechanics     Hybrid Journal   (Followers: 8)
Computational Methods and Function Theory     Hybrid Journal  
Computational Optimization and Applications     Hybrid Journal   (Followers: 10)
Computers & Mathematics with Applications     Full-text available via subscription   (Followers: 11)
Concrete Operators     Open Access   (Followers: 4)
Confluentes Mathematici     Hybrid Journal  
Constructive Mathematical Analysis     Open Access   (Followers: 1)
Contributions to Discrete Mathematics     Open Access   (Followers: 1)
Contributions to Game Theory and Management     Open Access  
COSMOS     Hybrid Journal  
Cryptography and Communications     Hybrid Journal   (Followers: 14)
Cuadernos de Investigación y Formación en Educación Matemática     Open Access  
Cubo. A Mathematical Journal     Open Access  
Current Research in Biostatistics     Open Access   (Followers: 8)
Czechoslovak Mathematical Journal     Hybrid Journal   (Followers: 1)
Daya Matematis : Jurnal Inovasi Pendidikan Matematika     Open Access   (Followers: 1)
Demographic Research     Open Access   (Followers: 15)

        1 2 3 4 5 | Last

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Communications in Combinatorics and Optimization
Number of Followers: 0  

  This is an Open Access Journal Open Access journal
ISSN (Print) 2538-2128 - ISSN (Online) 2538-2136
Published by Azarbaijan Shahide Madani University Homepage  [1 journal]
  • Signed total Italian k-domination in graphs

    • Abstract: Let k ≥ 1 be an integer, and let G be a finite and simple graph with vertex set V (G). A signed total Italian k-dominating function (STIkDF) on a graph G is a functionf : V (G) → {−1, 1, 2} satisfying the conditions that $sum_{xin N(v)}f(x)ge k$ for each vertex v ∈ V (G), where N(v) is the neighborhood of $v$, and each vertex u with f(u)=-1 is adjacent to a vertex v with f(v)=2 or to two vertices w and z with f(w)=f(z)=1. The weight of an STIkDF f is$omega(f)=sum_{vin V(G)}f(v)$. The signed total Italian k-domination number $gamma_{stI}^k(G)$ of G is the minimum weight of an STIkDF on G. In this paper we initiate the study of the signed total Italian k-dominationnumber of graphs, and we present different bounds on $gamma_{stI}^k(G)$. In addition, we determine thesigned total Italian k-domination number of some classes of graphs. Some of our results are extensions ofwell-known properties of the signed total Roman $k$-domination number $gamma_{stR}^k(G)$,introduced and investigated by Volkmann [9,12].
       
  • Þ-energy of generalized Petersen graphs

    • Abstract: ÞFor a given graph G, its Þ-energy is the sum of the absolute values of the eigenvalues of the Þ-matrix of G. In this article, we explore the Þ-energy of generalized Petersen graphs  G(p,k)  for various vertex partitions such as independent, domatic, total domatic and k-ply domatic partitions and partition containing a perfect matching in G(p,k). Further, we present a python program to obtain the Þ-energy of G(p,k)  for the vertex partitions under consideration and examine the relation between them.
       
  • Distinct edge geodetic decomposition in graphs

    • Abstract: Let G=(V,E) be a simple connected graph of order p and size q. A decomposition of a graph G is a collection π of edge-disjoint subgraphs G_1,G_2,…,G_n of G such that every edge of G belongs to exactly one G_i,(1≤i ≤n). The decomposition 〖π={G〗_1,G_2,…,G_n} of a connected graph G is said to be a distinct edge geodetic decomposition if g_1 (G_i )≠g_1 (G_j ),(1≤i≠j≤n). The maximum cardinality of π is called the distinct edge geodetic decomposition number of G and is denoted by π_dg1 (G), where g_1 (G) is the edge geodetic number of G. Some general properties satisfied by this concept are studied. Connected graphs which are edge geodetic decomposable are characterized. Connected distinct edge geodetic decomposable graphs of order p with π_dg1 (G)= p-2 are characterised.
       
  • Weak signed Roman k-domatic number of a graph

    • Abstract: Let k≥ 1  be an integer. A  weak signed Roman k-dominating function on a graph G isa function f:V (G)→ {-1, 1, 2} such that ΣuεN[v] f(u)≥ k for everyvε V(G), where N[v] is the closed neighborhood of v.A set {f1,f2, ... ,fd} of distinct weak signed Roman k-dominatingfunctions on G with the property that Σ1≤i≤d fi(v)≤ k for each vε V(G), is called a weak signed Roman k-dominating family (of functions) on G. The maximum number of functionsin a weak signed Roman k-dominating family on G is the  weak signed Roman k-domatic number} of Gdenoted by dwsR k(G). In this paper we initiate the study of the weak signed Roman $k$-domatic numberin graphs, and we present sharp bounds for dwsR k(G). In addition, we determine the weak signed Roman k-domatic number of some graphs.
       
  • A characterization relating domination, semitotal domination and total
           Roman domination in trees

    • Abstract: A total Roman dominating function on a graph $G$ is a function $f: V(G) rightarrow {0,1,2}$ such that for every vertex $vin V(G)$ with $f(v)=0$ there exists a vertex $uin V(G)$ adjacent to $v$ with $f(u)=2$, and the subgraph induced by the set ${xin V(G): f(x)geq 1}$ has no isolated vertices. The total Roman domination number of $G$, denoted $gamma_{tR}(G)$, is the minimum weight $omega(f)=sum_{vin V(G)}f(v)$ among all total Roman dominating functions $f$ on $G$.It is known that $gamma_{tR}(G)geq gamma_{t2}(G)+gamma(G)$ for any graph $G$ with neither isolated vertex nor components isomorphic to $K_2$, where $gamma_{t2}(G)$ and $gamma(G)$ represent the semitotal domination number and the classical domination number, respectively. In this paper we give a constructive characterization of the trees that satisfy the equality above.
       
  • A second-order corrector wide neighborhood infeasible interior-point
           method for linear optimization based on a specific kernel function

    • Abstract: In this paper, we present a second-order corrector infeasibleinterior-point method for linear optimization in a largeneighborhood of the central path. The innovation of our method is tocalculate the predictor directions using a specific kernel functioninstead of the logarithmic barrier function. We decompose thepredictor direction induced by the kernel function to two orthogonaldirections of the corresponding to the negative and positivecomponent of the right-hand side vector of the centering equation.The method then considers the new point as a linear combination ofthese directions along with a second-order corrector direction. Theconvergence analysis of the proposed method is investigated and itis proved that the complexity bound is Ο(n5/4 log ε-1).
       
  • Stirling number of the fourth kind and lucky partitions of a finite set

    • Abstract: The concept of Lucky k-polynomials and in particular Lucky χ-polynomials was recently introduced. This paper introduces Stirling number of the fourth kind and Lucky partitions of a finite set in order to determine either the Lucky k- or Lucky χ-polynomial of a graph. The integer partitions influence Stirling partitions of the second kind.
       
  • On the powers of signed graphs

    • Abstract: A signed graph is a graph in which each edge has a positive or negative sign. In this article, we define n^th power of a signed graph and discuss some properties of these powers of signed graphs. As we can define two types of signed graphs as the power of a signed graph, necessary and sufficient conditions are given for an n^th power of a signed graph to be unique. Also, we characterize balanced power signed graphs.
       
  • Two upper bounds on the A_α-spectral radius of a connected graph

    • Abstract: If A(G) and D(G) are respectively the adjacency matrix and the diagonal matrix of vertex degrees of a connected graph G, the generalized adjacency matrix Aα(G) is defined as Aα(G)=α D(G)+(1-α) A(G), where 0≤ α ≤ 1. The Aα (or generalized) spectral radius λ(Aα(G)) (or simply λα) is the largest eigenvalue of Aα(G). In this paper, we show that                                           λα ≤αΔ+(1-α)(2m(1-1/ω))1/2, where m, Δ and ω=ω(G) are respectively the size, the largest degree and the clique number of $G$. Further, if G has order n, then we show that                      2λα ≤ max1≤i≤n [αdi  + √α2 di ^2 +4mi(1-α)[α+(1-α)mj]  where di  and mi  are respectively the degree and the average 2-degree of the vertex vi.
       
  • Total outer-convex domination number of graphs

    • Abstract: In this paper, we initiate the study of total outer-convex domination as a new variant of graph domination and we show the close relationship that exists between this novel parameter and other domination parameters of a graph such as total domination, convex domination, and outer-convex domination. Furthermore, we obtain general bounds of total outer-convex domination number and, for some particular families of graphs, we obtain closed formulas.
       
  • The Tutte polynomial of matroids constructed by a family of splitting
           operations

    • Abstract: To extract some more information from the constructions of matroids that arise from new operations, computing the Tutte polynomial, plays an important role. In this paper, we consider applying three operations of splitting, element splitting and splitting off to a binary matroid and then introduce the Tutte polynomial of resulting matroids by these operations in terms of that of original matroids.
       
  • Total domination in cubic Knodel graphs

    • Abstract: A subset D of vertices of a graph G is a dominating set if for each u ∈ V (G) \ D, u is adjacent to somevertex v ∈ D. The domination number, γ(G) ofG, is the minimum cardinality of a dominating set of G. A setD ⊆ V (G) is a total dominating set if for eachu ∈ V (G), u is adjacent to some vertex v ∈ D. Thetotal domination number, γt (G) of G, is theminimum cardinality of a total dominating set of G. For an eveninteger $nge 2$ and $1\le Delta \le lfloorlog_2nrfloor$, aKnodel graph $W_{Delta,n}$ is a $Delta$-regularbipartite graph of even order n, with vertices (i,j), for$i=1,2$ and $0le jle n/2-1$, where for every $j$, $0le jlen/2-1$, there is an edge between vertex $(1, j)$ and every vertex$(2,(j+2^k-1)$ mod (n/2)), for $k=0,1,cdots,Delta-1$. In thispaper, we determine the total domination number in $3$-regularKnodel graphs $W_{3,n}$.
       
  • The annihilator-inclusion Ideal graph of a commutative ring

    • Abstract: Let R be a commutative ring with non-zero identity. The annihilator-inclusion ideal graph of R , denoted by ξR, is a graph whose vertex set is the of allnon-zero proper ideals of $R$ and two distinct vertices $I$ and $J$ are adjacentif and only if either Ann(I) ⊆ J or Ann(J) ⊆ I. In this paper, we investigate the basicproperties of the graph ξR. In particular, we showthat ξR is a connected graph with diameter at most three, andhas girth 3 or ∞. Furthermore, we determine all isomorphic classes of non-local Artinian rings whose annihilator-inclusion ideal graphs have genus zero or one.
       
  • On the variable sum exdeg index and cut edges of graphs

    • Abstract: The variable sum exdeg index of a graph G is defined as $SEI_a(G)=sum_{uin V(G)}d_G(u)a^{d_G(u)}$, where $aneq 1$ is a positive real number,  du(u) is the degree of a vertex u ∈ V (G). In this paper, we characterize the graphs with the extremum variable sum exdeg index among all the graphs having a fixed number of vertices and cut edges, for every a>1.
       
  • Some remarks on the sum of the inverse values of the normalized signless
           Laplacian eigenvalues of graphs

    • Abstract: Let G=(V,E), $V={v_1,v_2,ldots,v_n}$, be a simple connected graph with $%n$ vertices, $m$ edges and a sequence of vertex degrees $d_1geqd_2geqcdotsgeq d_n>0$, $d_i=d(v_i)$. Let ${A}=(a_{ij})_{ntimes n}$ and ${%D}=mathrm{diag }(d_1,d_2,ldots , d_n)$ be the adjacency and the diagonaldegree matrix of $G$, respectively. Denote by ${mathcal{L}^+}(G)={D}^{-1/2}(D+A) {D}^{-1/2}$ the normalized signless Laplacian matrix of graph $G$. Theeigenvalues of matrix $mathcal{L}^{+}(G)$, $2=gamma _{1}^{+}geq gamma_{2}^{+}geq cdots geq gamma _{n}^{+}geq 0$, are normalized signlessLaplacian eigenvalues of $G$. In this paper some bounds for the sum $%K^{+}(G)=sum_{i=1}^nfrac{1}{gamma _{i}^{+}}$ are considered.
       
  • Outer independent Roman domination number of trees

    • Abstract: ‎A Roman dominating function (RDF) on a graph G=(V,E) is a function  f : V → {0, 1, 2}  such that every vertex u for which f(u)=0 is‎ ‎adjacent to at least one vertex v for which f(v)=2‎. ‎An RDF f is called‎‎an outer independent Roman dominating function (OIRDF) if the set of‎‎vertices assigned a 0 under f is an independent set‎. ‎The weight of an‎‎OIRDF is the sum of its function values over all vertices‎, ‎and the outer‎‎independent Roman domination number ΥoiR (G) is the minimum weight‎‎of an OIRDF on $G$‎. ‎In this paper‎, ‎we show that if T is a tree of order n ≥ 3 with s(T) support vertices‎, ‎then $gamma _{oiR}(T)leq min‎ {‎frac{5n}{6},frac{3n+s(T)}{4}}.$ Moreover‎, ‎we characterize the tress‎‎attaining each bound‎.
       
  • Strength of strongest dominating sets in fuzzy graphs

    • Abstract: A set S of vertices in a graph G=(V,E) is a dominating set ofG if every vertex of V-S is adjacent to some vertex of S.For an integer k≥1, a set S of vertices is a k-step dominating set if any vertex of $G$ is at distance k from somevertex of S. In this paper, using membership values of vertices and edges in fuzzy graphs, we introduce the concepts of strength of strongestdominating set as well as strength of strongest$k$-step dominating set in fuzzy graphs. We determine various bounds forthese parameters in fuzzy graphs. We also determine the strengthof strongest dominating set in some families of fuzzy graphsincluding complete fuzzy graphs and complete bipartite fuzzygraphs.
       
  • Line completion number of grid graph Pn × Pm

    • Abstract: The concept of super line graph was introduced in the year 1995 by Bagga, Beineke and Varma. Given a graph with at least r edges, the super line graph of index r, Lr(G), has as its vertices the sets of r-edges of G, with two adjacent if there is an edge in one set adjacent to an edge in the other set. The line completion number lc(G) of a graph G is the least positive integer r for which Lr(G) is a complete graph. In this paper, we find the line completion number of grid graph Pn × Pm for various cases of n and m.
       
  • On the outer independent 2-rainbow domination number of Cartesian products
           of paths and cycles

    • Abstract: ‎Let G be a graph‎. ‎A 2-rainbow dominating function (or‎ 2-RDF) of G is a function f from V(G)‎ ‎to the set of all subsets of the set {1,2}‎ ‎such that for a vertex v ∈ V (G) with f(v) = ∅, ‎the‎‎condition $bigcup_{uin N_{G}(v)}f(u)={1,2}$ is fulfilled‎, wher NG(v)  is the open neighborhood‎‎of v‎. ‎The weight of 2-RDF f of G is the value‎‎$omega (f):=sum _{vin V(G)} f(v) $‎. ‎The 2-rainbow‎‎domination number of G‎, ‎denoted by Υr2 (G)‎, ‎is the‎‎minimum weight of a 2-RDF of G‎. ‎A 2-RDF f is called an outer independent 2-rainbow dominating function ‎(or OI2-RDF) of G if‎‎the set of all v ∈ V (G) with f(v) = ∅ is an‎ ‎independent set‎. ‎The outer independent 2-rainbow domination number Υoir2  (G) is‎‎the minimum weight of an OI2-RDF of G‎. ‎In this paper‎, ‎we obtain the‎‎outer independent 2-rainbow domination number of Pm□Pn‎ ‎and‎ Pm□Cn‎. ‎Also we determine the value of Υoir2  (Cm2Cn) when m or n is even‎.
       
  • New Bounds on the Energy of a Graph

    • Abstract: The energy of a graph G, denoted by Ε(G), is defined as the sum of the absolute values of all eigenvalues of G. In this paper, lower and upper bounds for energy in some of the graphs are established, in terms of graph invariants such as the number of vertices, the number of edges, and the number of closed walks.
       
  • A counterexample to a conjecture of Jafari Rad and Volkmann

    • Abstract: In this short note, we disprove the conjecture of Jafari Rad and Volkmann that every γ-vertex critical graph is  γR-vertex critical, where γ(G)  and γR(G) stand for the domination number and the Roman domination number of a graph G, respectively.
       
  • Algorithmic Aspects of Quasi-Total Roman Domination in Graphs

    • Abstract: For a simple, undirected, connected graph G=(V,E), a function f : V(G) →{0, 1, 2} which satisfies the following conditions is called a quasi-total Roman dominating function (QTRDF) of G with weight f(V(G))=ΣvΕV(G) f(v).C1). Every vertex uεV for which f(u) = 0 must be adjacent to at least one vertex v with f(v) = 2, and C2). Every vertex uεV for which f(u) = 2 must be adjacent to at least one vertex v with f(v)≥1.  For a graph G, the smallest possible weight of a QTRDF of G denoted γqtR(G) is known as the quasi-total Roman domination number of G. The problem of determining γqtR(G) of a graph G is called minimum quasi-total Roman domination problem (MQTRDP). In this paper, we show that the problem of determining whether G has a QTRDF of weight at most l is NP-complete for split graphs, star convex bipartite graphs, comb convex bipartite graphs and planar graphs. On the positive side, we show that MQTRDP for threshold graphs, chain graphs and bounded treewidth graphs is linear time solvable. Finally, an integer linear programming formulation for MQTRDP is presented.    
       
  • On the 2-independence subdivision number of graphs

    • Abstract: A subset S of vertices in a graph G = (V;E) is 2-independent if every vertexof S has at most one neighbor in S: The 2-independence number is the maximumcardinality of a 2-independent set of G: In this paper, we initiate the study of the2-independence subdivision number sdβ2(G) defined as the minimum numberof edges that must be subdivided (each edge in G can be subdivided at mostonce) in order to increase the 2-independence number. We first show that forevery connected graph G of order at least three, 1≤sdβ2(G)≤2; and we give anecessary and sufficient condition for graphs G attaining each bound. Moreover,restricted to the class of trees, we provide a constructive characterization of alltrees T with sdβ2(T)= 2; and we show that such a characterization suggestsan algorithm that determines whether a tree T has sdβ2(T)= 2 or sdβ2(T) = 1in polynomial time.    
       
  • A note on δ^(k)-colouring of the Cartesian product of some graphs

    • Abstract: The chromatic number, Χ(G) of a graph G is the minimum number of colours used in a proper colouring of G. In improper colouring, an edge uv is bad if the colours assigned to the end vertices of the edge is the same. Now, if the available colours are less than that of the chromatic number of graph G, then colouring the graph with the available colours leads to bad edges in G. The number of bad edges resulting from a δ(k)-colouring of G is denoted by bk(G). In this paper, we use the concept of δ(k)-colouring and determine the number of bad edges in the Cartesian product of some graphs.
       
 
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