Saeid Alikhani
Yazd University, Mathematics, Faculty Member
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Professor of Mathematics, Yazd University, Iranedit
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Let ) , ( = E V G be a simple graph. The Hosoya polynomial of G is ) , ( ) ( } , { = ) , ( v u d G V v u x x G H , where ) , ( v u d denotes the distance between vertices u and v . The dendrimer nanostar is a part of a new group of... more
Let ) , ( = E V G be a simple graph. The Hosoya polynomial of G is ) , ( ) ( } , { = ) , ( v u d G V v u x x G H , where ) , ( v u d denotes the distance between vertices u and v . The dendrimer nanostar is a part of a new group of macromolecules that seem photon funnels just like artificial antennas and also is a great resistant of photo bleaching. In this paper we compute the Hosoya polynomial of an infinite family of dendrimer nanostar denoted by ] [ 3 n D .
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Mathematics and Dendrimer
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Let $G=(V,E)$ be a simple graph. An independent dominating set of $G$ is a vertex subset that is both dominating and independent in $G$. The {\it independent domination polynomial} of a graph $G$ is the polynomial $D_i(G,x)=\sum_{A}... more
Let $G=(V,E)$ be a simple graph. An independent dominating set of $G$ is a vertex subset that is both dominating and independent in $G$. The {\it independent domination polynomial} of a graph $G$ is the polynomial $D_i(G,x)=\sum_{A} x^{|A|}$, summed over all independent dominating subsets $A\subseteq V$. A root of $D_i(G,x)$ is called an independence domination root. We prove that all independence domination roots of a claw-free graph are real. We consider clique cover product and present formula for the independent domination polynomial of this kind of product. As consequences, we construct graphs whose independence domination roots are real.
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... To My wife and my son Elahe and Amir Hossein For their great patience ... Oleh sebab masalah menentukan set dominasi dan bilangan set dominasi bagi sebarang graf adalah diketahui NP-lengkap, kita kaji polinomial dominasi bagi... more
... To My wife and my son Elahe and Amir Hossein For their great patience ... Oleh sebab masalah menentukan set dominasi dan bilangan set dominasi bagi sebarang graf adalah diketahui NP-lengkap, kita kaji polinomial dominasi bagi kelas-kelas graf dengan pembinaan tertentu. ...
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In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph G is a pair G = (V, E), where V and E are the vertex set and the edge set of G, respectively.... more
In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph G is a pair G = (V, E), where V and E are the vertex set and the edge set of G, respectively. The order and size of G is the number of vertices and edges of G, respectively. The degree or valency of a vertex u in a graph G (loopless), denoted by deg (u), is the number of edges meeting at u. If, for every vertex ν in G, deg (ν) = k, we say that G is a k-regular graph. The cycle of order n is denoted by Cn and is a connected 2-regular graph. The path graph of order n is denoted by Pn and obtain by deleting an edge of Cn. A tree is an undirected graph in which any two vertices are connected by exactly one path, or equivalently a connected undirected graph without cycle. A leaf (or pendant vertex) of a tree is a vertex of the tree of degree 1. An edge of a graph is said to be pendant if one of its vertices is a pendant vertex. A complete bipart...
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In [2] a new method (which is called MZ-algorithm), has presented for dividing a natural number x by two and used graphs as models to show MZ-algorithm. Every digit in a number denoted by a vertex and edges of graph draw based on... more
In [2] a new method (which is called MZ-algorithm), has presented for dividing a natural number x by two and used graphs as models to show MZ-algorithm. Every digit in a number denoted by a vertex and edges of graph draw based on MZ-algorithm. We have shown the division of the number 458 by two in Figure 1. is graph (Figure 1) which we call it division graph by two (DGBT) is a path of order 13, i.e., P13 (see [2]). Applying k-times of the MZmethod for the number x, creates a graph with unique structure that is denoted byGk(x) and is called DGBT. It is easy to see that Gk(n) is not tree for k > 1, since the graph has cycle. See the graph G2(375) in Figure 2 (see [2]). Since DGBT is an innite graph, sometimes is beer to show some of DGBT in bitmap model. In bitmap model each numbers in 0, 1, 2, ..., 9 represented by unique color. See Figure 3.
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Let G be a simple graph of order n. The energy E(G) of G is the sum of the absolute values of the eigenvalues of G. The Randi´cRandi´c matrix of G, denoted by R(G), is defined as the n×n matrix whose (i, j)-entry is (didj) −1 2 if vi and... more
Let G be a simple graph of order n. The energy E(G) of G is the sum of the absolute values of the eigenvalues of G. The Randi´cRandi´c matrix of G, denoted by R(G), is defined as the n×n matrix whose (i, j)-entry is (didj) −1 2 if vi and vj are adjacent and 0 for another cases. The Randi´cRandi´c energy RE(G) of G is the sum of absolute values of the eigen-values of R(G). In this paper we compute the energy and the Randi´cRandi´c energy for certain graphs. We also propose a conjecture on the Randi´cRandi´c energy.
The domination polynomial of a graph G of order n is the polynomial D(G, x) = P n i=γ(G) d(G, i)x i , where d(G, i) is the number of dominating sets of G of size i, and γ(G) is the domination number of G. In this paper, we obtain some... more
The domination polynomial of a graph G of order n is the polynomial D(G, x) = P n i=γ(G) d(G, i)x i , where d(G, i) is the number of dominating sets of G of size i, and γ(G) is the domination number of G. In this paper, we obtain some properties of the coefficients of D(G, x). Also, by study of the dominating sets and the domination polynomials of specific graphs denoted by G ′ (m), we obtain a relationship between the domination polynomial of graphs containing an induced path of length at least three, and the domination polynomial of related graphs obtained by replacing the path by shorter path. As examples of graphs G ′ (m), we study the dominating sets and domination polynomials of cycles and generalized theta graphs. Finally, we show that, if n ≡ 0, 2(mod 3) and D(G, x) = D(Cn, x), then G = Cn.
The distinguishing number D(G) of a graph G is the least integer d such that G has a vertex labeling with d labels that is preserved only by a trivial automorphism. The distinguishing stability, of a graph G is denoted by st_D(G) and is... more
The distinguishing number D(G) of a graph G is the least integer d such that G has a vertex labeling with d labels that is preserved only by a trivial automorphism. The distinguishing stability, of a graph G is denoted by st_D(G) and is the minimum number of vertices whose removal changes the distinguishing number. We obtain a general upper bound st_D(G) ≤ V(G) -D(G)+1, and a relationships between the distinguishing stabilities of graphs G and G-v, i.e., st_D(G)≤ st_D(G-v)+1, where v∈ V(G). Also we study the edge distinguishing stability number (distinguishing bondage number) of G.
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Combinatorics and Graph
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Let $G=(V,E)$ be a simple graph. A set $I\subseteq V$ is an independent set, if no two of its members are adjacent in $G$. The $k$-independent graph of $G$, $I_k (G)$, is defined to be the graph whose vertices correspond to the... more
Let $G=(V,E)$ be a simple graph. A set $I\subseteq V$ is an independent set, if no two of its members are adjacent in $G$. The $k$-independent graph of $G$, $I_k (G)$, is defined to be the graph whose vertices correspond to the independent sets of $G$ that have cardinality at most $k$. Two vertices in $I_k(G)$ are adjacent if and only if the corresponding independent sets of $G$ differ by either adding or deleting a single vertex. In this paper, we obtain some properties of $I_k(G)$ and compute it for some graphs.