 # TRANSACTIONS ON COMBINATORICSYear:2014 | Volume:3 | Issue:4

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Year

Volume(Issue)

Issues        Writer:
Issue Info:
• Year:

2014
• Volume:

3
• Issue:

4
• Start Page:

1
• End Page:

9
Keywords:
Abstract:

Let G be a simple graph with vertex set V (G) ={v1, v2,… vn} and edge set E (G) = {e1, e2,… em}. Similar to the Randic matrix, here we introduce the Randic incidence matrix of a graph G, denoted by IR (G), which is defined as the n×m matrix whose (I, j) -entry is (di) -1/2 if vi is incident to ej and 0 otherwise. Naturally, the Randic incidence energy IRE of G is the sum of the singular values of IR (G). We establish lower and upper bounds for the Randic incidence energy. Graphs for which these bounds are best possible are characterized. Moreover, we investigate the relation between the Randic incidence energy of a graph and that of its subgraphs. Also we give a sharp upper bound for the Randic incidence energy of a bipartite graph and determine the trees with the maximum Randic incidence energy among all n-vertex trees. As a result, some results are very different from those for incidence energy.

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Title:
Writer:
Issue Info:
• Year:

2014
• Volume:

3
• Issue:

4
• Start Page:

11
• End Page:

18
Keywords:
Abstract:

A signed graph (marked graph) is an ordered pair S= (G, s) (S= (G, m)), where G= (V, E) is a graph called the underlying graph of S and s: E ® {+,-} (m: V® {+,-}) is a function. For a graph G, V (G), E (G) and C (G) denote its vertex set, edge set and cut-vertex set, respectively. The lict graph Lc (G) of a graph G= (V, E) is defined as the graph having vertex set E (G) È C (G) in which two vertices are adjacent if and only if they correspond to adjacent edges of G or one corresponds to an edge ei of G and the other corresponds to a cut-vertex cj of G such that ei is incident with cj. In this paper, we introduce lict sigraphs, as a natural extension of the notion of lict graph to the realm of signed graphs. We show that every lict sigraph is balanced. We characterize signed graphs S and S' for which S ~ Lc (S), h (S) ~ Lc (S), L (S) ~ Lc (S'), J (S) ~ Lc (S') and T1 (S) ~ Lc (S'), where h (S), L (S), J (S) and T1 (S) are negation, line graph, jump graph and semitotal line sigraph of S, respectively, and ~ means switching equivalence.

Yearly Impact: View 6558 Download 5211 Citation 0 Refrence 0
Writer:
Issue Info:
• Year:

2014
• Volume:

3
• Issue:

4
• Start Page:

19
• End Page:

30
Keywords:
Abstract:

A subset S of vertices in a graph G is called a geodetic set if every vertex not in S lies on a shortest path between two vertices from S. A subset D of vertices in G is called dominating set if every vertex not in D has at least one neighbor in D. A geodetic dominating set S is both a geodetic and a dominating set. The geodetic (domination, geodetic domination) number g(G) (g(G), gg (G)) of G is the minimum cardinality among all geodetic (dominating, geodetic dominating) sets in G. In this paper, we show that if a triangle free graph G has minimum degree at least 2 and g (G) =2, then gg(G) = g(G). It is shown, for every nontrivial connected graph G with g (G) =2 and diam (G)>3, that gg (G)>g (G). The lower bound for the geodetic domination number of Cartesian product graphs is proved. Geodetic domination number of product of cycles (paths) are determined. In this work, we also determine some bounds and exact values of the geodetic domination number of strong product of graphs.

Yearly Impact: View 11987 Download 7371 Citation 0 Refrence 0
Writer:
Issue Info:
• Year:

2014
• Volume:

3
• Issue:

4
• Start Page:

31
• End Page:

41
Keywords:
Abstract:

Please click on PDF to view the abstract.

Yearly Impact: View 6563 Download 5369 Citation 0 Refrence 0
Writer:
Issue Info:
• Year:

2014
• Volume:

3
• Issue:

4
• Start Page:

43
• End Page:

54
Keywords:
Abstract:

In this work, using eigenvalues and eigenvectors of unitary Cayley graphs over finite local rings and elementary linear algebra, we characterize which local rings allow a PST occurring in its unitary Cayley graph. Moreover, we have some developments when R is a product of local rings.

Yearly Impact: View 10105 Download 3863 Citation 0 Refrence 0
Writer:
Issue Info:
• Year:

2014
• Volume:

3
• Issue:

4
• Start Page:

55
• End Page:

58
Keywords:
Abstract:

Please click on PDF to view the abstract.

Yearly Impact: View 7291 Download 3303 Citation 0 Refrence 0