On the Edge Metric Dimension of Certain Polyphenyl Chains

The most productive application of graph theory in chemistry is the representation of molecules by the graphs, where vertices and edges of graphs are the atoms and valence bonds between a pair of atoms, respectively. For a vertex w and an edge f = c 1 c 2 of a connected graph G , the minimum number from distances of w with c 1 and c 2 is called the distance between w and f . If for every two distinct edges f 1 , f 2 ∈ E G , there always exists w 1 ∈ W E ⊆ V G such that d f 1 , w 1 ≠ d f 2 , w 1 , then W E is named as an edge metric generator. The minimum number of vertices in W E is known as the edge metric dimension of G . In this paper, we calculate the edge metric dimension of ortho-polyphenyl chain graph O n , meta-polyphenyl chain graph M n , and the linear [n]-tetracene graph T n and also find the edge metric dimension of para-polyphenyl chain graph L n . It has been proved that the edge metric dimension of O n , M n , and T n is bounded, while L n is unbounded.

Financial Risk Analysis for SMEs with Graph-based Supply Chain Mining

Small and Medium-sized Enterprises (SMEs) are playing a vital role in the modern economy. Recent years, financial risk analysis for SMEs attracts lots of attentions from financial institutions. However, the financial risk analysis for SMEs usually suffers data deficiency problem, especially for the mobile financial institutions which seldom collect credit-related data directly from SMEs. Fortunately, although credit-related information of SMEs is hard to be acquired sufficiently, the interactive relationships between SMEs, which may contain valuable information of financial risk, is usually available for the mobile financial institutions. Finding out credit-related relationship of SME from massive interactions helps comprehensively model the SMEs thus improve the performance of financial risk analysis. In this paper, tackling the data deficiency problem of financial risk analysis for SMEs, we propose an innovative financial risk analysis framework with graph-based supply chain mining. Specifically, to capture the credit-related topology structural and temporal variation information of SMEs, we design and employ a novel spatial-temporal aware graph neural network, to mine supply chain relationship on a SME graph, and then analysis the credit risk based on the mined supply chain graph. Experimental results on real-world financial datasets prove the effectiveness of our proposal for financial risk analysis for SMEs.

Computing total edge irregularity strength of some n-uniform cactus chain graphs and related chain graphs

<div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p><span>Given graph </span><em>G</em><span>(</span><span><em>V</em>,<em>E</em></span><span>)</span><span>. We use the notion of total </span><em>k</em><span>-labeling which is edge irregular. The notion </span>of total edge irregularity strength (tes) of graph <em>G</em> means the minimum integer <em>k</em> used in the edge irregular total k-labeling of <em>G</em>. A cactus graph <em>G</em> is a connected graph where no edge lies in more than one cycle. A cactus graph consisting of some blocks where each block is cycle <em>C_n</em> with same size <em>n</em> is named an <em>n</em>-uniform cactus graph. If each cycle of the cactus graph has no more than two cut-vertices and each cut-vertex is shared by exactly two cycles, then <em>G</em> is called <em>n</em>-uniform cactus chain graph. In this paper, we determine tes of n-uniform cactus chain graphs <em>C</em>(<em>C_n^r</em>) of length <em>r</em> for some <em>n</em> ≡ 0 mod 3. We also investigate tes of related chain graphs, i.e. tadpole chain graphs <em>T_r</em>(4,<em>n</em>) and <em>T_r</em>(5,<em>n</em>) of length <em>r</em>. Our results are as follows: tes(<em>C</em>(<em>C_n^r</em>)) = ⌈(<em>nr</em> + 2)/3⌉ ; tes(<em>T_r</em>(4,<em>n</em>)) = ⌈((5+<em>n</em>)<em>r</em>+2)/3⌉ ; tes(<em>T_r</em>(5,<em>n</em>)) = ⌈((5+<em>n</em>)<em>r</em>+2)/3⌉.</p></div></div></div>