Relationship between adjacency and distance matrix of graph of diameter two

The relationship among every pair of vertices in a graph can be represented as a matrix, such as in adjacency matrix and distance matrix. Both adjacency and distance matrices have the same property. Adjacency and distance matrices are both symmetric matrix with diagonals entries equals to 0. In this paper, we discuss relationships between adjacency matrix and distance matrix of a graph of diameter two, which is <em>D=2(J-I)-A</em>. From this relationship, we also determine the value of the determinant matrix <em>A+D</em> and the upper bound of determinant of matrix <em>D</em>.

On the Locating Chromatic Number of Barbell Shadow Path Graph

The locating-chromatic number was introduced by Chartrand in 2002. The locating chromatic number of a graph is a combined concept between the coloring and partition dimension of a graph. The locating chromatic number of a graph is defined as the cardinality of the minimum color classes of the graph. In this paper, we discuss about the locating-chromatic number of shadow path graph and barbell graph containing shadow graph.

On local antimagic vertex coloring of corona products related to friendship and fan graph

Let <em>G</em>=(<em>V</em>,<em>E</em>) be connected graph. A bijection <em>f </em>: <em>E</em> → {1,2,3,..., |<em>E</em>|} is a local antimagic of <em>G</em> if any adjacent vertices <em>u,v</em> ∈ <em>V</em> satisfies <em>w</em>(<em>u</em>)≠ <em>w</em>(<em>v</em>), where <em>w</em>(<em>u</em>)=∑_e∈E(u)<em>f</em>(<em>e</em>), <em>E</em>(<em>u</em>) is the set of edges incident to <em>u</em>. When vertex <em>u</em> is assigned the color <em>w</em>(<em>u</em>), we called it a local antimagic vertex coloring of <em>G</em>. A local antimagic chromatic number of <em>G</em>, denoted by <em>χ</em>_la(<em>G</em>), is the minimum number of colors taken over all colorings induced by the local antimagic labeling of <em>G</em>. In this paper, we determine the local antimagic chromatic number of corona product of friendship and fan with null graph on <em>m</em> vertices, namely, <em>χ</em>_la(<em>F</em>_n ⊙ \overline{K_m}) and <em>χ</em>_la(<em>f</em>_(1,n) ⊙ \overline{K_m}).

The degree sequences of a graph with restrictions

<p>Two finite sequences <em>s</em>₁and <em>s</em>₂ of nonnegative integers are called bigraphical if there exists a bipartite graph <em>G</em> with partite sets <em>V</em>₁ and <em>V</em>₂ such that <em>s</em>₁ and <em>s</em>₂ are the degrees in <em>G </em>of the vertices in <em>V</em>₁ and <em>V</em>₂, respectively. In this paper, we introduce the concept of <em>1</em>-graphical sequences and present a necessary and sufficient condition for a sequence to be <em>1</em>-graphical in terms of bigraphical sequences.</p>

Structure of intersection graphs

<p style="text-align: left;" dir="ltr"> Let <em>G</em> be a finite group and let <em>N</em> be a fixed normal subgroup of <em>G</em>. In this paper, a new kind of graph on <em>G</em>, namely the intersection graph is defined and studied. We use <img src="/public/site/images/ikhsan/equation.png" alt="" width="6" height="4" /> to denote this graph, with its vertices are all normal subgroups of <em>G</em> and two distinct vertices are adjacent if their intersection in <em>N</em>. We show some properties of this graph. For instance, the intersection graph is a simple connected with diameter at most two. Furthermore we give the graph structure of <img src="/public/site/images/ikhsan/equation_(1).png" alt="" width="6" height="4" /> for some finite groups such as the symmetric, dihedral, special linear group, quaternion and cyclic groups. </p>

All unicyclic graphs of order n with locating-chromatic number n-3

<p class="p1">Characterizing all graphs having a certain locating-chromatic number is not an easy task. In this paper, we are going to pay attention on finding all unicyclic graphs of order <em>n</em> (⩾ 6) and having locating-chromatic number <em>n</em>-3.</p>

Odd Harmonious Labeling of Pn ⊵ C4 and Pn ⊵ D2(C4)

A graph <em>G</em> with <em>q</em> edges is said to be odd harmonious if there exists an injection <em>f</em>:<em>V</em>(<em>G</em>) → ℤ_2q so that the induced function <em>f</em>*:<em>E</em>(<em>G</em>)→ {1,3,...,2<em>q</em>-1} defined by <em>f</em>*(<em>uv</em>)=<em>f</em>(<em>u</em>)+<em>f</em>(<em>v</em>) is a bijection.<p>Here we show that graphs constructed by edge comb product of path <em>P</em>_n and cycle on four vertices <em>C</em>₄ or shadow of cycle of order four <em>D</em>₂(<em>C</em>₄) are odd harmonious.</p>

On Super (a,d)-edge antimagic total labeling of branched-prism graph

Let <em>H</em> be a branched-prism graph, denoted by <em>H</em> = (<em>C_m</em> x <em>P</em>₂) ⊙ Ǩ_n for odd <em>m</em>, <em>m</em> ≥ 3 and <em>n</em> ≥ 1. This paper considers about the existence of the super (<em>a</em>,<em>d</em>)-edge antimagic total labeling of <em>H</em>, for some positive integer <em>a</em> and some non-negative integer <em>d</em>.

Total edge irregularity strength of some cycle related graphs

<p>An edge irregular total <em>k</em>-labeling <em>f</em> : <em>V</em> ∪ <em>E</em> → 1,2, ..., <em>k</em> of a graph <em>G</em> = (<em>V,E</em>) is a labeling of vertices and edges of <em>G</em> in such a way that for any two different edges <em>uv</em> and <em>u'v'</em>, their weights <em>f</em>(<em>u</em>)+<em>f</em>(<em>uv</em>)+<em>f</em>(<em>v</em>) and <em>f</em>(<em>u'</em>)+<em>f</em>(<em>u'v'</em>)+<em>f</em>(<em>v'</em>) are distinct. The total edge irregularity strength tes(<em>G</em>) is defined as the minimum <em>k</em> for which the graph <em>G</em> has an edge irregular total <em>k</em>-labeling. In this paper, we determine the total edge irregularity strength of new classes of graphs <em>C_m</em> @ <em>C_n</em>, <em>P_m,n</em>* and <em>C_m,n</em>* and hence we extend the validity of the conjecture tes(<em>G</em>) = max {⌈|<em>E</em>(<em>G</em>)|+2)/3⌉, ⌈(Δ(<em>G</em>)+1)/2⌉}<em> </em> for some more graphs.</p>

Computing the split domination number of grid graphs

<p>A set <em>D</em> - <em>V</em> is a dominating set of <em>G</em> if every vertex in <em>V - D</em> is adjacent to some vertex in <em>D</em>. The dominating number γ(<em>G</em>) of <em>G</em> is the minimum cardinality of a dominating set <em>D</em>. A dominating set <em>D</em> of a graph <em>G</em> = (<em>V;E</em>) is a split dominating set if the induced graph (<em>V</em> - <em>D</em>) is disconnected. The split domination number γ<em>_s</em>(<em>G</em>) is the minimum cardinality of a split domination set. In this paper we have introduced a new method to obtain the split domination number of grid graphs by partitioning the vertex set in terms of star graphs and also we have<br />obtained the exact values of γ<em>s</em>(<em>G_m;n</em>); <em>m</em> ≤ <em>n</em>; <em>m,n</em> ≤ 24:</p>