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MATHEMATICS (714 journals)            First | 1 2 3 4     

Showing 601 - 538 of 538 Journals sorted alphabetically
Research in the Mathematical Sciences     Open Access  
Research Journal of Pure Algebra     Open Access   (Followers: 1)
Researches in Mathematics     Open Access  
Results in Control and Optimization     Open Access  
Results in Mathematics     Hybrid Journal  
Results in Nonlinear Analysis     Open Access  
Review of Symbolic Logic     Full-text available via subscription   (Followers: 2)
Reviews in Mathematical Physics     Hybrid Journal   (Followers: 1)
Revista Baiana de Educação Matemática     Open Access  
Revista Bases de la Ciencia     Open Access  
Revista BoEM - Boletim online de Educação Matemática     Open Access  
Revista Colombiana de Matemáticas     Open Access   (Followers: 1)
Revista de Ciencias     Open Access  
Revista de Educación Matemática     Open Access  
Revista de la Escuela de Perfeccionamiento en Investigación Operativa     Open Access  
Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas     Partially Free  
Revista de Matemática : Teoría y Aplicaciones     Open Access   (Followers: 1)
Revista Digital: Matemática, Educación e Internet     Open Access  
Revista Electrónica de Conocimientos, Saberes y Prácticas     Open Access  
Revista Integración : Temas de Matemáticas     Open Access  
Revista Internacional de Sistemas     Open Access  
Revista Latinoamericana de Etnomatemática     Open Access  
Revista Latinoamericana de Investigación en Matemática Educativa     Open Access  
Revista Matemática Complutense     Hybrid Journal  
Revista REAMEC : Rede Amazônica de Educação em Ciências e Matemática     Open Access  
Revista SIGMA     Open Access  
Ricerche di Matematica     Hybrid Journal  
RMS : Research in Mathematics & Statistics     Open Access  
Royal Society Open Science     Open Access   (Followers: 7)
Russian Journal of Mathematical Physics     Full-text available via subscription  
Russian Mathematics     Hybrid Journal  
Sahand Communications in Mathematical Analysis     Open Access  
Sampling Theory, Signal Processing, and Data Analysis     Hybrid Journal  
São Paulo Journal of Mathematical Sciences     Hybrid Journal  
Science China Mathematics     Hybrid Journal   (Followers: 1)
Science Progress     Full-text available via subscription   (Followers: 1)
Sciences & Technologie A : sciences exactes     Open Access  
Selecta Mathematica     Hybrid Journal   (Followers: 1)
SeMA Journal     Hybrid Journal  
Semigroup Forum     Hybrid Journal   (Followers: 1)
Set-Valued and Variational Analysis     Hybrid Journal  
SIAM Journal on Applied Mathematics     Hybrid Journal   (Followers: 11)
SIAM Journal on Computing     Hybrid Journal   (Followers: 11)
SIAM Journal on Control and Optimization     Hybrid Journal   (Followers: 18)
SIAM Journal on Discrete Mathematics     Hybrid Journal   (Followers: 8)
SIAM Journal on Financial Mathematics     Hybrid Journal   (Followers: 3)
SIAM Journal on Mathematics of Data Science     Hybrid Journal   (Followers: 1)
SIAM Journal on Matrix Analysis and Applications     Hybrid Journal   (Followers: 3)
SIAM Journal on Optimization     Hybrid Journal   (Followers: 12)
Siberian Advances in Mathematics     Hybrid Journal  
Siberian Mathematical Journal     Hybrid Journal  
Sigmae     Open Access  
SILICON     Hybrid Journal  
SN Partial Differential Equations and Applications     Hybrid Journal  
Soft Computing     Hybrid Journal   (Followers: 7)
Statistics and Computing     Hybrid Journal   (Followers: 14)
Stochastic Analysis and Applications     Hybrid Journal   (Followers: 3)
Stochastic Partial Differential Equations : Analysis and Computations     Hybrid Journal   (Followers: 2)
Stochastic Processes and their Applications     Hybrid Journal   (Followers: 6)
Stochastics and Dynamics     Hybrid Journal   (Followers: 2)
Studia Scientiarum Mathematicarum Hungarica     Full-text available via subscription   (Followers: 1)
Studia Universitatis Babeș-Bolyai Informatica     Open Access  
Studies In Applied Mathematics     Hybrid Journal   (Followers: 1)
Studies in Mathematical Sciences     Open Access   (Followers: 1)
Superficies y vacio     Open Access  
Suska Journal of Mathematics Education     Open Access   (Followers: 1)
Swiss Journal of Geosciences     Hybrid Journal   (Followers: 1)
Synthesis Lectures on Algorithms and Software in Engineering     Full-text available via subscription   (Followers: 2)
Synthesis Lectures on Mathematics and Statistics     Full-text available via subscription   (Followers: 1)
Tamkang Journal of Mathematics     Open Access  
Tatra Mountains Mathematical Publications     Open Access  
Teaching Mathematics     Full-text available via subscription   (Followers: 10)
Teaching Mathematics and its Applications: An International Journal of the IMA     Hybrid Journal   (Followers: 4)
Teaching Statistics     Hybrid Journal   (Followers: 8)
Technometrics     Full-text available via subscription   (Followers: 8)
The Journal of Supercomputing     Hybrid Journal   (Followers: 1)
The Mathematica journal     Open Access  
The Mathematical Gazette     Full-text available via subscription   (Followers: 1)
The Mathematical Intelligencer     Hybrid Journal  
The Ramanujan Journal     Hybrid Journal  
The VLDB Journal     Hybrid Journal   (Followers: 2)
Theoretical and Mathematical Physics     Hybrid Journal   (Followers: 7)
Theory and Applications of Graphs     Open Access  
Topological Methods in Nonlinear Analysis     Full-text available via subscription  
Transactions of the London Mathematical Society     Open Access   (Followers: 1)
Transformation Groups     Hybrid Journal  
Turkish Journal of Mathematics     Open Access  
Ukrainian Mathematical Journal     Hybrid Journal  
Uniciencia     Open Access  
Uniform Distribution Theory     Open Access  
Unisda Journal of Mathematics and Computer Science     Open Access  
Unnes Journal of Mathematics     Open Access   (Followers: 1)
Unnes Journal of Mathematics Education     Open Access   (Followers: 2)
Unnes Journal of Mathematics Education Research     Open Access   (Followers: 1)
Ural Mathematical Journal     Open Access  
Vestnik Samarskogo Gosudarstvennogo Tekhnicheskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki     Open Access  
Vestnik St. Petersburg University: Mathematics     Hybrid Journal  
VFAST Transactions on Mathematics     Open Access   (Followers: 1)
Vietnam Journal of Mathematics     Hybrid Journal  
Vinculum     Full-text available via subscription  
Visnyk of V. N. Karazin Kharkiv National University. Ser. Mathematics, Applied Mathematics and Mechanics     Open Access   (Followers: 2)
Water SA     Open Access   (Followers: 1)
Water Waves     Hybrid Journal  
Zamm-Zeitschrift Fuer Angewandte Mathematik Und Mechanik     Hybrid Journal   (Followers: 1)
ZDM     Hybrid Journal   (Followers: 2)
Zeitschrift für angewandte Mathematik und Physik     Hybrid Journal   (Followers: 2)
Zeitschrift fur Energiewirtschaft     Hybrid Journal  
Zetetike     Open Access  

  First | 1 2 3 4     

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Theory and Applications of Graphs
Number of Followers: 0  

  This is an Open Access Journal Open Access journal
ISSN (Online) 2470-9859
Published by Georgia Southern University Homepage  [10 journals]
  • Radio Number of Hamming Graphs of Diameter 3

    • Authors: Jason DeVito et al.
      Abstract: For $G$ a simple, connected graph, a vertex labeling $f:V(G)\to \Z_+$ is called a \emph{radio labeling of $G$} if it satisfies $ f(u)-f(v) \geq\diam(G)+1-d(u,v)$ for all distinct vertices $u,v\in V(G)$. The \emph{radio number of $G$} is the minimal span over all radio labelings of $G$. If a bijective radio labeling onto $\{1,2,\dots, V(G) \}$ exists, $G$ is called a \emph{radio graceful} graph. We determine the radio number of all diameter 3 Hamming graphs and show that an infinite subset of them is radio graceful.
      PubDate: Thu, 25 Aug 2022 04:41:08 PDT
       
  • Restrained reinforcement number in graphs

    • Authors: Kazhal Haghparast et al.
      Abstract: A set $S$ of vertices is a restrained dominating set of a graph $G=(V,E)$ if every vertex in $V\setminus S$ has a neighbor in $S$ and a neighbor in $V\setminus S$. The minimum cardinality of a restrained dominating set is the restrained domination number $\gamma_{r}(G)$. In this paper we initiate the study of the restrained reinforcement number $r_{r}(G)$ of a graph $G$ defined as the cardinality of a smallest set of edges $F\subseteq E(\overline{G})$ for which $\gamma _{r}(G+F)
      PubDate: Wed, 24 Aug 2022 04:31:31 PDT
       
  • On the Integer-antimagic Spectra of Non-Hamiltonian Graphs

    • Authors: Wai Chee Shiu et al.
      Abstract: Let $A$ be a nontrivial abelian group. A connected simple graph $G = (V, E)$ is $A$-\textbf{antimagic} if there exists an edge labeling $f: E(G) \to A \setminus \{0\}$ such that the induced vertex labeling $f^+: V(G) \to A$, defined by $f^+(v) = \Sigma$ $\{f(u,v): (u, v) \in E(G) \}$, is a one-to-one map. In this paper, we analyze the group-antimagic property for Cartesian products, hexagonal nets and theta graphs.
      PubDate: Wed, 24 Aug 2022 04:31:20 PDT
       
  • Total Colouring of New Classes of Subcubic graphs

    • Authors: Sethuraman G et al.
      Abstract: The total chromatic number of a graph $G$, denoted $\chi^{\prime\prime}(G)$, is the least number of colours needed to colour the vertices and the edges of $G$ such that no incident or adjacent elements (vertices or edges) receive the same colour. The popular Total Colouring Conjecture (TCC) posed by Behzad states that, for every simple graph $G$, $\chi^{\prime\prime}(G) \leq \Delta(G)+2$. In this paper, we prove that the total chromatic number for a family of subcubic graphs called cube connected paths and also for a class of subcubic graphs having the property that the vertices are covered by independent triangles are exactly $\Delta(G)+1$. More precisely, these two families of subcubic graphs are shown to be Type 1 graph.\\
      PubDate: Wed, 24 Aug 2022 04:31:10 PDT
       
  • Geodesic bipancyclicity of the Cartesian product of graphs

    • Authors: Amruta V. Shinde et al.
      Abstract: A cycle containing a shortest path between two vertices $u$ and $v$ in a graph $G$ is called a $(u,v)$-geodesic cycle. A connected graph $G$ is geodesic 2-bipancyclic, if every pair of vertices $u,v$ of it is contained in a $(u,v)$-geodesic cycle of length $l$ for each even integer $l$ satisfying $2d + 2\leq l \leq V(G) ,$ where $d$ is the distance between $u$ and $v.$ In this paper, we prove that the Cartesian product of two geodesic hamiltonian graphs is a geodesic 2-bipancyclic graph. As a consequence, we show that for $n \geq 2$ every $n$-dimensional torus is a geodesic 2-bipancyclic graph.
      PubDate: Mon, 18 Jul 2022 14:06:16 PDT
       
  • On the Total Set Chromatic Number of Graphs

    • Authors: Mark Anthony C. Tolentino et al.
      Abstract: Given a vertex coloring c of a graph, the neighborhood color set of a vertex is defined to be the set of all of its neighbors’ colors. The coloring c is called a set coloring if any two adjacent vertices have different neighborhood color sets. The set chromatic number χs(G) of a graph G is the minimum number of colors required in a set coloring of G. In this work, we investigate a total analog of set colorings; that is, we study set colorings of the total graph of graphs. Given a graph G = (V, E), its total graph T(G) is the graph whose vertex set is V ∪ E and in which two vertices are adjacent if and only if their corresponding elements in G are adjacent or incident. First, we establish sharp bounds for the set chromatic number of the total graph of a graph. Furthermore, we study the set colorings of the total graph of different families of graphs.
      PubDate: Mon, 18 Jul 2022 14:06:05 PDT
       
  • Harmonious Labelings Via Cosets and Subcosets

    • Authors: Jared L. Painter et al.
      Abstract: In \cite{AR}, it is shown that the disjoint union of an odd cycle and certain paths is harmonious, and that certain starlike trees are harmonious using properties of cosets for a particular subgroup of $\mathbb{Z}_m$, where $m$ is the number of edges of the graph. We expand upon these results by first exploring the numerical properties when adding values from cosets and subcosets in $\mathbb{Z}_m$. We will then show that these properties may be used to harmoniously label graphs involving a more complex starlike tree, which we will call the snowflake graph.
      PubDate: Mon, 18 Jul 2022 14:01:21 PDT
       
  • On $p$-competition graphs of loopless Hamiltonian digraphs without
           symmetric arcs and graph operations

    • Authors: Kuniharu Yokomura et al.
      Abstract: For a digraph $D$, the $p$-competition graph $C_{p}(D)$ of $D$ is the graph satisfying the following: $V(C_{p}(D))=V(D)$, for $x,y \in V(C_{p}(D))$, $xy \in E(C_{p}(D))$ if and only if there exist distinct $p$ vertices $v_{1},$ $v_{2},$ $...,$ $v_{p}$ $\in$ $V(D)$ such that $x \rightarrow v_{i}$, $y \rightarrow v_{i}$ $\in$ $A(D)$ for each $i=1,2,$ $...,$ $p$.We show the $H_1 \cup H_2$ is a $p$-competition graph of a loopless digraph without symmetric arcs for $p \geq 2$, where $H_1$ and $H_2$ are $p$-competition graphs of loopless digraphs without symmetric arcs and $V(H_1) \cap V(H_2)$ $=$ $\{ \alpha \}$. For $p$-competition graphs of loopless Hamiltonian digraphs without symmetric arcs, we obtain similar results. And we show that a star $K_{1,n}$ is a $p$-competition graph of a loopless Hamiltonian digraph without symmetric arcs if $n \geq 2p+3$ and $p \geq 3$.Based on these results, we obtain conditions such that spiders, caterpillars and cacti are $p$-competition graphs of loopless digraphs without symmetric arcs. We also obtain conditions such that these graphs are $p$-competition graphs of loopless Hamiltonian digraphs without symmetric arcs.
      PubDate: Mon, 18 Jul 2022 14:01:10 PDT
       
  • One-factorizations of the complete graph $K_{p+1}$ arising from parabolas

    • Authors: György Kiss et al.
      Abstract: There are three types of affine regular polygons in AG(2, q): ellipse, hyperbola and parabola. The first two cases have been investigated in previous papers. In this note, a particular class of geometric one-factorizations of the complete graph Kn arising from parabolas is constructed and described in full detail. With the support of computer aided investigation, it is also conjectured that up to isomorphisms this is the only one-factorization where each one-factor is either represented by a line or a parabola.
      PubDate: Wed, 06 Jul 2022 06:51:26 PDT
       
  • Characterization of outerplanar graphs with equal 2-domination and
           domination numbers

    • Authors: Naoki Matsumoto
      Abstract: A {\em $k$-domination number} of a graph $G$ is minimum cardinality of a $k$-dominating set of $G$, where a subset $S \subseteq V(G)$ is a {\em $k$-dominating set} if each vertex $v\in V(G)\setminus S$ is adjacent to at least $k$ vertices in $S$. It is known that for any graph $G$ with $\Delta(G) \geq k \geq 2$, $\gamma_k(G) \geq \gamma(G) + k - 2$, and then $\gamma_k(G)> \gamma(G)$ for any $k\geq 3$, where $\gamma(G) = \gamma_1(G)$ is the usual domination number. Thus, it is the most interesting problem to characterize graphs $G$ with $\gamma_2(G) = \gamma(G)$. In this paper, we characterize outerplanar graphs with equal 2-domination and domination numbers.
      PubDate: Wed, 06 Jul 2022 06:51:17 PDT
       
  • Rainbow Perfect and Near-Perfect Matchings in Complete Graphs with Edges
           Colored by Circular Distance

    • Authors: Shuhei Saitoh et al.
      Abstract: Given an edge-colored complete graph Kn on n vertices, a perfect (respectively, near-perfect) matching M in Kn with an even (respectively, odd) number of vertices is rainbow if all edges have distinct colors. In this paper, we consider an edge coloring of Kn by circular distance, and we denote the resulting complete graph by K●n. We show that when K●n has an even number of vertices, it contains a rainbow perfect matching if and only if n=8k or n=8k+2, where k is a nonnegative integer. In the case of an odd number of vertices, Kirkman matching is known to be a rainbow near-perfect matching in K●n. However, real-world applications sometimes require multiple rainbow near-perfect matchings. We propose a method for using a recursive algorithm to generate multiple rainbow near-perfect matchings in K●n.
      PubDate: Sat, 25 Jun 2022 04:01:11 PDT
       
  • Ultrametrics and Complete Multipartite Graphs

    • Authors: Viktoriia Viktorivna Bilet et al.
      Abstract: Let \((X, d)\) be a semimetric space and let \(G\) be a graph. We say that \(G\) is the diametrical graph of \((X, d)\) if \(X\) is the vertex set of \(G\) and the adjacency of vertices \(x\) and \(y\) is equivalent to the equality \(\diam X = d(x, y)\). It is shown that a semimetric space \((X, d)\) with diameter \(d^*\) is ultrametric iff the diametrical graph of \((X, d_{\varepsilon})\) with \(d_{\varepsilon}(x, y) = \min\{d(x, y), \varepsilon\}\) is complete multipartite for every \(\varepsilon \in (0, d^*)\). A refinement of the last result is given for totally bounded ultrametric spaces. Moreover, using complete multipartite graphs we characterize the compact ultrametrizable topological spaces. The bounded ultrametric spaces, which are weakly similar to unbounded ones, are also characterized via complete multipartite graphs.
      PubDate: Sat, 25 Jun 2022 03:51:13 PDT
       
  • An Even 2-Factor in the Line Graph of a Cubic Graph

    • Authors: SeungJae Eom et al.
      Abstract: An even 2-factor is one such that each cycle is of even length. A 4- regular graph G is 4-edge-colorable if and only if G has two edge-disjoint even 2- factors whose union contains all edges in G. It is known that the line graph of a cubic graph without 3-edge-coloring is not 4-edge-colorable. Hence, we are interested in whether those graphs have an even 2-factor. Bonisoli and Bonvicini proved that the line graph of a connected cubic graph G with an even number of edges has an even 2-factor, if G has a perfect matching [Even cycles and even 2-factors in the line graph of a simple graph, Electron. J. Combin. 24 (2017), P4.15]. In this paper, we extend this theorem to the line graph of a connected cubic graph G satisfying certain conditions.
      PubDate: Thu, 19 May 2022 11:56:05 PDT
       
  • Prime labelings on planar grid graphs

    • Authors: Stephen James Curran
      Abstract: It is known that for any prime p and any integer n such that 1≤n≤p there exists a prime labeling on the pxn planar grid graph PpxPn. We show that PpxPn has a prime labeling for any odd prime p and any integer n such that that p
      PubDate: Thu, 24 Mar 2022 07:06:37 PDT
       
  • Characterizing Edge Betweenness-Uniform graphs

    • Authors: Jana Coroničová Hurajová et al.
      Abstract: The \emph{betweenness centality} of an edge $e$ is, summed over all $u,v\in V(G)$, the ratio of the number of shortest $u,v$-paths in $G$ containing $e$ to the number of shortest $u,v$-paths in $G$. Graphs whose vertices all have the same edge betweenness centrality are called \emph{edge betweeness-uniform}. It was recently shown by Madaras, Hurajová, Newman, Miranda, Fl\'{o}rez, and Narayan that of the over 11.7 million graphs with ten vertices or fewer, only four graphs are edge betweenness-uniform but not edge-transitive.In this paper we present new results involving properties of betweenness-uniform graphs.
      PubDate: Thu, 24 Mar 2022 07:06:27 PDT
       
  • Chromatic Polynomials of Signed Book Graphs

    • Authors: Deepak Sehrawat et al.
      Abstract: For $m \geq 3$ and $n \geq 1$, the $m$-cycle \textit{book graph} $B(m,n)$ consists of $n$ copies of the cycle $C_m$ with one common edge. In this paper, we prove that (a) the number of switching non-isomorphic signed $B(m,n)$ is $n+1$, and (b) the chromatic number of a signed $B(m,n)$ is either 2 or 3. We also obtain explicit formulas for the chromatic polynomials and the zero-free chromatic polynomials of switching non-isomorphic signed book graphs.
      PubDate: Thu, 24 Mar 2022 07:06:16 PDT
       
  • Application of the Combinatorial Nullstellensatz to Integer-magic Graph
           Labelings

    • Authors: Richard M. Low et al.
      Abstract: Let $A$ be a nontrivial abelian group and $A^* = A \setminus \{0\}$. A graph is $A$-magic if there exists an edge labeling $f$ using elements of $A^*$ which induces a constant vertex labeling of the graph. Such a labeling $f$ is called an $A$-magic labeling and the constant value of the induced vertex labeling is called an $A$-magic value. In this paper, we use the Combinatorial Nullstellensatz to show the existence of $\mathbb{Z}_p$-magic labelings (prime $p \geq 3$ ) for various graphs, without having to construct the $\mathbb{Z}_p$-magic labelings. Through many examples, we illustrate the usefulness (and limitations) in applying the Combinatorial Nullstellensatz to the integer-magic labeling problem. Finally, we focus on $\mathbb{Z}_3$-magic labelings and give some results for various classes of graphs.
      PubDate: Tue, 22 Feb 2022 14:46:32 PST
       
  • Connectedness of Unit Distance Subgraphs Induced by Closed Convex Sets

    • Authors: Remie Janssen et al.
      Abstract: The unit distance graph $G^1_{R^d}$ is the infinite graph whose nodes are points in $R^d$, with an edge between two points if the Euclidean distance between these points is $1$. The 2-dimensional version $G^1_{R^2}$ of this graph is typically studied for its chromatic number, as in the Hadwiger-Nelson problem. However, other properties of unit distance graphs are rarely studied. Here, we consider the restriction of $G^1_{R^d}$ to closed convex subsets $X$ of $R^d$. We show that the graph $G^1_{R^d}[X]$ is connected precisely when the radius of $r(X)$ of $X$ is equal to $0$, or when $r(X)\geq 1$ and the affine dimension of $X$ is at least $2$. For hyperrectangles, we give bounds for the graph diameter in the critical case that the radius is exactly 1.
      PubDate: Tue, 22 Feb 2022 14:41:33 PST
       
  • Facial Achromatic Number of Triangulations with Given Guarding Number

    • Authors: Naoki Matsumoto et al.
      Abstract: A (not necessarily proper) $k$-coloring $c : V(G) \rightarrow \{1,2,\dots,k\}$ of a graph $G$ on a surface is a {\em facial $t$-complete $k$-coloring} if every $t$-tuple of colors appears on the boundary of some face of $G$. The maximum number $k$ such that $G$ has a facial $t$-complete $k$-coloring is called a {\em facial $t$-achromatic number} of $G$, denoted by $\psi_t(G)$. In this paper, we investigate the relation between the facial 3-achromatic number and guarding number of triangulations on a surface, where a {\em guarding number} of a graph $G$ embedded on a surface, denoted by $\gd(G)$, is the smallest size of its {\em guarding set} which is a generalized concept of guards in the art gallery problem. We show that for any graph $G$ embedded on a surface, $\psi_{\Delta(G^*)}(G) \leq \gd(G) + \Delta(G^*) - 1$, where $\Delta(G^*)$ is the largest face size of $G$. Furthermore, we investigate sufficient conditions for a triangulation $G$ on a surface to satisfy $\psi_{3}(G) = \gd(G) + 2$. In particular, we prove that every triangulation $G$ on the sphere with $\gd(G) = 2$ satisfies the above equality and that for one with guarding number $3$, it also satisfies the above equality with sufficiently large number of vertices.
      PubDate: Wed, 16 Feb 2022 09:26:17 PST
       
  • A Complete Characterisation of Vertex-multiplications of Trees with
           Diameter 5

    • Authors: Willie Han Wah Wong et al.
      Abstract: For a connected graph $G$, let $\mathscr{D}(G)$ be the family of strong orientations of $G$; and for any $D\in\mathscr{D}(G)$, we denote by $d(D)$ the diameter of $D$. The $\textit{orientation number}$ of $G$ is defined as $\bar{d}(G)=\min\{d(D)\mid D\in \mathscr{D}(G)\}$. In 2000, Koh and Tay introduced a new family of graphs, $G$ vertex-multiplications, and extended the results on the orientation number of complete $n$-partite graphs. Suppose $G$ has the vertex set $V(G)=\{v_1,v_2,\ldots, v_n\}$. For any sequence of $n$ positive integers $(s_i)$, a $G$ \textit{vertex-multiplication}, denoted by $G(s_1, s_2,\ldots, s_n)$, is the graph with vertex set $V^*=\bigcup_{i=1}^n{V_i}$ and edge set $E^*$, where $V_i$'s are pairwise disjoint sets with $ V_i =s_i$, for $i=1,2,\ldots,n$; and for any $u,v\in V^*$, $uv\in E^*$ if and only if $u\in V_i$ and $v\in V_j$ for some $i,j\in \{1,2,\ldots, n\}$ with $i\neq j$ such that $v_i v_j\in E(G)$. They proved a fundamental classification of $G$ vertex-multiplications, with $s_i\ge 2$ for all $i=1,2,\ldots, n$, into three classes $\mathscr{C}_0, \mathscr{C}_1$ and $\mathscr{C}_2$, and any vertex-multiplication of a tree with diameter at least 3 does not belong to the class $\mathscr{C}_2$. Furthermore, some necessary and sufficient conditions for $\mathscr{C}_0$ were established for vertex-multiplications of trees with diameter $5$. In this paper, we give a complete characterisation of vertex-multiplications of trees with diameter $5$ in $\mathscr{C}_0$ and $\mathscr{C}_1$.
      PubDate: Mon, 15 Nov 2021 10:31:27 PST
       
 
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