Randi'c incidence energy of graphs
Let $G$ be a simple graph with vertex set $V(G) = {v_1, v_2,ldots , v_n}$ and edge set $E(G) = {e_1, e_2,ldots , e_m}$. Similar to the Randi'c matrix, here we introduce the Randi'c incidence matrix of a graph $G$, denoted by $I_R(G)$, which is defined as the $ntimes m$ matrix whose $(i...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
University of Isfahan
2014-12-01
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| Series: | Transactions on Combinatorics |
| Subjects: | |
| Online Access: | http://www.combinatorics.ir/pdf_5573_68f2261c2087d1f09fb34c2f8de4b053.html |
| Summary: | Let $G$ be a simple graph with vertex set $V(G) = {v_1, v_2,ldots
, v_n}$ and edge set $E(G) = {e_1, e_2,ldots , e_m}$. Similar to
the Randi'c matrix, here we introduce the Randi'c incidence matrix
of a graph $G$, denoted by $I_R(G)$, which is defined as the
$ntimes m$ matrix whose $(i, j)$-entry is $(d_i)^{-frac{1}{2}}$ if
$v_i$ is incident to $e_j$ and $0$ otherwise. Naturally, the
Randi'c incidence energy $I_RE$ of $G$ is the sum of the singular
values of $I_R(G)$. We establish lower and upper bounds for the
Randi'c incidence energy. Graphs for which these bounds are best
possible are characterized. Moreover, we investigate the relation
between the Randi'c incidence energy of a graph and that of its
subgraphs. Also we give a sharp upper bound for the Randi'c
incidence energy of a bipartite graph and determine the trees with
the maximum Randi'c incidence energy among all $n$-vertex trees. As
a result, some results are very different from those for incidence
energy. |
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| ISSN: | 2251-8657 2251-8665 |