On (4,2)-digraph Containing a Cycle of Length 2

A diregular digraph is a digraph with the in-degree and out-degree of all vertices is constant. The Moore bound for a diregular digraph of degree d and diameter k is M_{d,k}=l+d+d^2+...+d^k. It is well known that diregular digraphs of order M_{d,k}, degree d>l tnd diameter k>l do not exist . A (d,k) -digraph is a diregular digraph of degree d>1, diameter k>1, and number of vertices one less than the Moore bound. For degrees d=2 and 3,it has been shown that for diameter k >= 3 there are no such (d,k)-digraphs. However for diameter 2, it is known that (d,2)-digraphs do exist for any degree d. The line digraph of K_{d+1} is one example of such (42)-digraphs. Furthermore, the recent study showed that there are three non-isomorphic(2,2)-digraphs and exactly one non-isomorphic (3,2)-digraph. In this paper, we shall study (4,2)-digraphs. We show that if (4,2)-digraph G contains a cycle of length 2 then G must be the line digraph of a complete digraph K_5

Determining Finite Connected Graphs Along the Quadratic Embedding Constants of Paths

The QE constant of a finite connected graph $G$, denoted by $\mathrm{QEC}(G)$, is by definition the maximum of the quadratic function associated to the distance matrix on a certain sphere of codimension two. We prove that the QE constants of paths $P_n$ form a strictly increasing sequence converging to $-1/2$. Then we formulate the problem of determining all the graphs $G$ satisfying $\mathrm{QEC}(P_n)\le\mathrm{QEC}(G)<\mathrm{QEC}(P_{n+1})$. The answer is given for $n=2$ and $n=3$ by exploiting forbidden subgraphs for $\mathrm{QEC}(G)<-1/2$ and the explicit QE constants of star products of the complete graphs.Comment: 24 pages, 6 figure

On the Metric Dimension of Corona Product of Graphs

For an ordered set W = {w_1,w_2,...,w_k} of vertices and a vertex v in a connected graph G, the representation of v with respect to W is the ordered k-tuple r(v|W) = (d(v,w_1),d(v,w_2),...,d(v,w_k)) where d(x,y) represents the distance between the vertices x and y. The set W is called a resolving set for G if every vertex of G has a distinct representation. A resolving set containing a minimum number of vertices is called a basis for G. The metric dimension of G, denoted by dim(G), is the number of vertices in a basis of G. A graph G corona H, G ⊙ H, is de�fined as a graph which formed by taking n copies of graphs H_1,H_2,...,H_n of H and connecting i-th vertex of G to the vertices of H_i. In this paper, we determine the metric dimension of corona product graphs G⊙H, the lower bound of the metric dimension of K_1 +H and determine some exact values of the metric dimension of G⊙H for some particular graphs H

The Uniqueness of Almost Moore Digraphs with Degree 4 And Diameter 2

Abstract. It is well known that Moore digraphs of degree d &gt; 1 and diameter k &gt; 1 do not exist. For degrees 2 and 3, it has been shown that for diameter k ≥ 3 there are no almost Moore digraphs, i.e. the diregular digraphs of order one less than the Moore bound. Digraphs with order close to the Moore bound arise in the construction of optimal networks. For diameter 2, it is known that almost Moore digraphs exist for any degree because the line digraphs of complete digraphs are examples of such digraphs. However, it is not known whether these are the only almost Moore digraphs. It is shown that for degree 3, there are no almost Moore digraphs of diameter 2 other than the line digraph of K4. In this paper, we shall consider the almost Moore digraphs of diameter 2 and degree 4. We prove that there is exactly one such digraph, namely the line digraph of K5. Ketunggalan Graf Berarah Hampir Moore dengan Derajat 4 dan Diameter 2Sari. Telah lama diketahui bahwa tidak ada graf berarah Moore dengan derajat d&gt;1 dan diameter k&gt;1. Lebih lanjut, untuk derajat 2 dan 3, telah ditunjukkan bahwa untuk diameter t&gt;3, tidak ada graf berarah Hampir Moore, yakni graf berarah teratur dengan orde satu lebih kecil dari batas Moore. Graf berarah dengan orde mendekati batas Moore digunakan dalam pcngkonstruksian jaringan optimal. Untuk diameter 2, diketahui bahwa graf berarah Hampir Moore ada untuk setiap derajat karena graf berarah garis (line digraph) dari graf komplit adalah salah satu contoh dari graf berarah tersebut. Akan tetapi, belum dapat dibuktikan apakah graf berarah tersebut merupakan satu-satunya contoh dari graf berarah Hampir Moore tadi. Selanjutnya telah ditunjukkan bahwa untuk derajat 3, tidak ada graf berarah Hampir Moore diameter 2 selain graf berarah garis dari K4. Pada makalah ini, kita mengkaji graf berarah Hampir Moore diameter 2 dan derajat 4. Kita buktikan bahwa ada tepat satu graf berarah tersebut, yaitu graf berarah garis dari K5

The Metric Dimension of Amalgamation of Cycles

For an ordered set W = {w_1, w_2 , ..., w_k } of vertices and a vertex v in a connected graph G, the representation of v with respect to W is the ordered k-tuple r(v|W) = (d(v,w_1), d(v,w_2 ), ..., d (v,w_k )), where d(x,y) represents the distance between the vertices x and y. The set W is called a resolving set for G if every vertex of G has a distinct representation. A resolving set containing a minimum number of vertices is called a basis for G. The dimension of G, denoted by dim(G), is the number of vertices in a basis of G. Let {G_i} be a finite collection of graphs and each G_i has a fixed vertex voi called a terminal. The amalgamation Amal {Gi , v_{oi}} is formed by taking all of the G_i’s and identifying their terminals. In this paper, we determine the metric dimension of amalgamation of cycles

The Resolving Graph of Amalgamation of Cycles

For an ordered set W = {w_1,w_2,...,w_k} of vertices and a vertex v in a connected graph G, the representation of v with respect to W is the ordered k-tuple r(v|W) = (d(v,w_1),d(v,w_2),...,d(v,w_k)) where d(x,y) represents the distance between the vertices x and y. The set W is called a resolving set for G if every vertex of G has a distinct representation. A resolving set containing a minimum number of vertices is called a basis for G. The dimension of G, denoted by dim(G), is the number of vertices in a basis of G. A resolving set W of G is connected if the subgraph induced by W is a nontrivial connected subgraph of G. The connected resolving number is the minimum cardinality of a connected resolving set in a graph G, denoted by cr(G). A cr-set of G is a connected resolving set with cardinality cr(G). A connected graph H is a resolving graph if there is a graph G with a cr-set W such that = H. Let {G_i} be a finite collection of graphs and each G_i has a fixed vertex v_{oi} called a terminal. The amalgamation Amal{Gi,v_{oi}} is formed by taking of all the G_i's and identifying their terminals. In this paper, we determine the connected resolving number and characterize the resolving graphs of amalgamation of cycles

The Metric Dimension of Graph with Pendant Edges

For an ordered set W = {w_1,w_2,...,w_k} of vertices and a vertex v in a connected graph G, the representation of v with respect to W is the ordered k-tuple r(v|W) = (d(v,w_1), d(v,w_2),..., d(v,w_k)) where d(x,y) represents the distance between the vertices x and y. The set W is called a resolving set for G if every two vertices of G have distinct representations. A resolving set containing a minimum number of vertices is called a basis for G. The dimension of G, denoted by dim(G), is the number of vertices in a basis of G. In this paper, we determine the dimensions of some corona graphs G⊙K_1, and G⊙K_m for any graph G and m ≥ 2, and a graph with pendant edges more general than corona graphs G⊙K_m

Total Vertex-Irregularity Labelings for Subdivision of Several Classes of Trees

AbstractMotivated by the notion of the irregularity strength of a graph introduced by Chartrand et al. [3] in 1988 and various kind of other total labelings, Baca et al. [1] introduced the total vertex irregularity strength of a graph.In 2010, Nurdin, Baskoro, Salman and Gaos[5] determined the total vertex irregularity strength for various types of trees, namely complete k–ary trees, a subdivision of stars, and subdivision of particular types of caterpillars. In other paper[6], they conjectured that the total vertex irregularity strength of any tree T is only determined by the number of vertices of degree 1, 2, and 3 in T . In this paper, we attempt to verify this conjecture by considering a subdivision of several types of trees, namely caterpillars, firecrackers, and amalgamation of stars

Buletin BSNP : Perguruan Tinggi BHMN Pasca Putusan Mahkamah Konstitusi

Di antara aspek yang memicu kon- tra adalah dugaan bahwa UU BHP me- lakukan penyeragaman bentuk dan tata kelola. Padahal UU BHP mengatur bahwa badan hukum pendidikan terdiri atas beragam bentuk badan hukum pendidikan yaitu BHP Pemerintah, BHP Masyarakat, BHP Penyelenggara (ya- yasan atau perkumpulan), dan BHP Pe- merintah Daerah, yang masing-masing memiliki tata kelola yang berbeda satu dengan yang lainnya