| Summary: | As an analogous concept of a nowhere-zero flow for directed graphs, zero-sum flows and constant-sum flows are defined and studied in the literature. For an undirected graph, a zero-sum flow (constant-sum flow resp.) is an assignment of nonzero integers to the edges such that the sum of the values of all edges incident with each vertex is zero (constant <i>h</i> resp.), and we call it a zero-sum <i>k</i>-flow (<i>h</i>-sum <i>k</i>-flow resp.) if the values of the edges are less than <i>k</i>. We extend these concepts to general constant-sum <i>A</i>-flow, where <i>A</i> is an Abelian group, and consider the case <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mo>=</mo><msub><mi mathvariant="double-struck">Z</mi><mi>k</mi></msub></mrow></semantics></math></inline-formula> the additive Abelian cyclic group of integer congruences modulo <i>k</i> with identity 0. In the literature, a graph is alternatively called <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><mi>k</mi></msub></semantics></math></inline-formula>-magic if it admits a constant-sum <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><mi>k</mi></msub></semantics></math></inline-formula>-flow, where the constant sum is called a magic sum or an index for short. We define the set of all possible magic sums such that <i>G</i> admits a constant-sum <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><mi>k</mi></msub></semantics></math></inline-formula>-flow to be <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>I</mi><mi>k</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and call it the magic sum spectrum, or for short, the index set of <i>G</i> with respect to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><mi>k</mi></msub></semantics></math></inline-formula>. In this article, we study the general properties of the magic sum spectrum of graphs. We determine the magic sum spectrum of complete bipartite graphs <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>K</mi><mrow><mi>m</mi><mo>,</mo><mi>n</mi></mrow></msub></semantics></math></inline-formula> for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>≥</mo><mi>n</mi><mo>≥</mo><mn>2</mn></mrow></semantics></math></inline-formula> as the additive cyclic subgroups of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><mi>k</mi></msub></semantics></math></inline-formula> generated by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfrac><mi>k</mi><mi>d</mi></mfrac></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><mo>=</mo><mi>g</mi><mi>c</mi><mi>d</mi><mo>(</mo><mi>m</mi><mo>−</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></semantics></math></inline-formula>. Also, we show that every regular graph <i>G</i> with a perfect matching has a full magic sum spectrum, namely, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>I</mi><mi>k</mi></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><msub><mi mathvariant="double-struck">Z</mi><mi>k</mi></msub></mrow></semantics></math></inline-formula> for all <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>≥</mo><mn>3</mn></mrow></semantics></math></inline-formula>. We characterize a 3-regular graph so that it admits a perfect matching if and only if it has a full magic sum spectrum, while an example is given for a 3-regular graph without a perfect matching which has no full magic sum spectrum. Another example is given for a 5-regular graph without a perfect matching, which, however, has a full magic sum spectrum. In particular, we completely determine the magic sum spectra for all regular graphs of even degree. As a byproduct, we verify a conjecture raised by Akbari et al., which claims that every connected <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>4</mn><mi>k</mi></mrow></semantics></math></inline-formula>-regular graph of even order admits a 1-sum 4-flow. More open problems are included.
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